Long Chord Method for Setting out simple Curve

Setting out simple Curve by long chord method requires some data to be calculated like sub chord length, long chord length, curve length, tangential angle, etc., and after calculation peg intervals are marked and with the help of instrument like total station points of the curve are plotted.

Long Chord Method

calculate the nesssary data for setting out simple curve using the long chord method.

Use the following data –

Radius (R) = 100cm

the through chainage of I = 621.995cm

Deflection angle (Φ)=101.32°

Peg interval = 10cm

calculation necessary data requires for long chord method

Tangent length (IT) = R*tan(Φ/2)

=100*tan(101.32/2) = 122cm

through chainage of T = chainage of I-IT

= 622-122 = 500cm

Curve length =RΦ(π/180)

Lc = 100*101.32(π/180) = 176.833cm

through chainage of U = chainage of T+Lc

= 500+176.83 = 676.83cm

Initial sub-chord =(the chainage of following point) – through chainage of T

following point = chainage of T+peg interval

= 500+10 = 510cm

Initial sub-chord

= 510-500 = 10cm

Final sub-chord = Lc-(the chainage prevoius point)

pevoius point = 170

Final sub-chord

= 176.83-170 = 6.83cm

No. genreal sub-chord =(Lc-initial sub-chord – final sub-chord)/peg interval)

No. = (176.83-10-6.83)/10 = 16

Tangential angle (t) = (chord length/R)*(90/π)

Initial sub-chord = 10cm

Tangential angle(t1) =(10/R)*(90/π)

= (10/100)*(90/π) = 2.864°

general sub-chord =10cm

Tangential angle(t1-t17) =(10/R)*(90/π) = 2.864°

Final sub-chord = 6.83cm

Tangential angle(t1-t17) =(6.83/100)*(90/π) = 1.956°

To set out the Centre line of the Road using the long chord method, the lengths of the chords must be calculated.

Chord length calculation

C1 = 2Rsin(t1) =(2*100)*sin(2.864) = 9.997cm

C2 = 2Rsin(t1+t2) =(2*100)*sin(2.864+2.864) = 19.961cm

C3 = 2Rsin(t1+(2t2)) =(2*100)sin(2.864+(2*2.864)) = 29.879cm

C4 = 2Rsin(t1+(3t2)) =(2*100)sin(2.864+(3*2.864)) = 39.720cm

C5 = 2Rsin(t1+(4t2)) =(2*100)sin(2.864+(4*2.864)) = 49.46cm

C6 = 2Rsin(t1+(5t2)) =(2*100)sin(2.864+(5*2.864)) = 59.088cm

C7 = 2Rsin(t1+(6t2)) =(2*100)sin(2.864+(6*2.864)) = 68.56cm

C8 = 2Rsin(t1+(7t2)) =(2*100)sin(2.864+(7*2.864)) = 77.863cm

C9 = 2Rsin(t1+(8t2)) =(2*100)sin(2.864+(8*2.864)) = 86.97cm

C10 = 2Rsin(t1+(9t2)) =(2*100)sin(2.864+(9*2.864)) = 95.86cm

C11 = 2Rsin(t1+(10t2)) =(2*100)sin(2.864+(10*2.864)) = 104.511cm

C12 = 2Rsin(t1+(11t2)) =(2*100)sin(2.864+(11*2.864)) = 112.901cm

C13 = 2Rsin(t1+(12t2)) =(2*100)sin(2.864+(12*2.864)) = 121.008cm

C14 = 2Rsin(t1+(13t2)) =(2*100)sin(2.864+(13*2.864)) = 128.814cm

C15 = 2Rsin(t1+(14t2)) =(2*100)sin(2.864+(14*2.864)) = 136.297cm

C16 = 2Rsin(t1+(15t2)) =(2*100)sin(2.864+(15*2.864)) = 143.440cm

C17 = 2Rsin(t1+(16t2)) =(2*100)sin(2.864+(16*2.864)) = 150.225cm

CU = 2Rsin(t1+(16t2)+1.956) =(2*100)sin(2.864+(16*2.864)+1.956) = 154.644cm

The last value should match with chord length = 2Rsin(Φ/2) = 154.679cm

Tangential Angle and Chord Distance Table

Point being set outChainage (m)Tangential angle chord to chord (Degree)Cummulative Tangential Angle from Tangent to chord (Degree)Chord Distance From point T(m)
T500009.997
C15102.8642.8649.997
C25202.8645.72819.961
C35302.8648.59229.879
C45402.86411.45639.72
C55502.86414.3249.46
C65602.86417.18459.088
C75702.86420.04868.56
C85802.86422.19277.863
C95902.86425.77686.97
C106002.86428.6495.86
C116102.86431.504104.511
C126202.86434.368112.901
C136302.86437.232121.008
C146402.86440.096128.814
C156502.86442.96136.297
C166602.86445.824143.44
C176702.86448.688150.225
CU676.831.95650.644154.644

To cross check, the last angle should be half of Deflection angle (Φ/2)

CU = 50.644

Φ = 101.32°

Φ = 101.32/2 = 50.644°

curve setting out procedure by long chord method

  • Fix the total station at point T.
  • Put the prism at point I.
  • Sight the prism on point I by total station.
  • Set the horizontal angle to zero.
  • To the fisrt chord point(C1) rotate the total station by given angle (2.864).
  • Measure the given chord Distance (9.97) and put the peg1 on this point.
  • To set second chord point(C2) rotate the total station by given angle (2.864).
  • Measure the given chord Distance (19.961) and put the peg2 on this point.
  • Repeat the same procedure for each chord point upto CU.
  • Cross check the mid ordinate,chord length,tangent length,external distance and curve length for verification.

calculation of other components of the curve

Mid – ordinate = R(1-cos(Φ/2))

= 100(1-cos(101.32/2)) = 36.608cm

External Distance = R(sec(Φ/2)-1)

= 100(sec(101.32/2)-1) = 57.748cm

Chord Length = 2Rsin(Φ/2)

= (2*100)*sin(101.32/2) = 154.68cm

Tangent Length = Rtan(Φ/2)

= 100tan(101.32/2) = 122cm

Curve Length (Lc) = R*Φ *(π/180)

= 100*101.32*(π/180) = 176.83cm

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