Section 3. Radial Chord Method

a. Circlular Geometry

For any circular arc, the angle between the tangent at one end of the arc and the chord is half the arc's central angle, Figure C-21.

 horiz70
Figure C-21 Deflection angle

 

 Angle a/2 is the deflection angle from one end of the arc to the other. The chord's length is computed from:

horiz72

     Equation C-10

 

In terms of the degree of curvature, Figure C-22: 

horiz71
Figure C-22 Full station deflection angle

 

The deflection angle for a full station is half the degree of curvature. Since the deflection angle is D/2 and it occurs over a 100.00 ft, the deflection rate can be computed from:

 horiz73         Equation C-11

 

Extending this geometry to the entire curve, Figure C-23, the total deflection angle at the BC from the PI to the EC is Δ/2.

horiz71b
Figure C-23 Deflection angle for entire curve

 

Since the deflection angle occurs across the curve's length, the deflection rate can also be written as Equation C-12.

 horiz73b         Equation C-12

 

b. Radial chords

One way to stake a horizontal curve is by the radial chord method, Figure C-24.

 horiz78
Figure C-24 Radial chord method

 

An instrument is set up on the BC and the PI sighted as a backsight. Then to stake each curve point, a defelction angle is turned and chord distance measured. For each point to be staked, we need to compute its deflection angle and chord distance.

The deflection angle to any point i on the curve, Figure C-25, can be computed from Equation C-13.

horiz74 
Figure C-25 Deflection angle and radial chord

  

horiz75        Equation C-13

 

li is the arc distance to the point from the BC and is computed using Equation C-14.

horiz76        Equation C-14

 

Equation C-10 can be re-written using the deflection angle:

 

horiz77         Equation C-15

 

Using Equations C-13 to C-15, the defelction angle and distance any curve point from the BC can be computed.

 

c. Example

Determine the radial chord stakeout data at full stations for the example from Section 2.d.

Summary of given and computed curve data:

Δ = 55°00'00" R = 500.00 ft  
D = 11°27'33.0" L = 479.965 ft T = 260.284 ft
LC = 461.749 ft E = 63.691 ft M = 56.494 ft

 

Point Station
PI 25+00.00
BC 22+39.716
EC 27+19.681 Bk = 27+60.284 Ah

horiz79

Use Equation C-11 to compute the curve's deflection rate:

horiz80

Set up Equations C-13, C-14, and C-15 for the curve:

horiz81

Set up the Curve Table: 

     Curve Point

Arc dist, li, (ft)

Defl angle,δi

Radial chord, ci

EC      27+19.681 Bk

 

 

 

       27+00

 

 

 

       26+00

 

 

 

       25+00

 

 

 

       24+00

 

 

 

       23+00

 

 

 

BC      22+39.716      

 

Solve the three equations for each curve point and record the results in the table.

At 22+39.716, we're still at the BC so all three entries are zero.

At 23+00:

horiz82

And so on until the table is complete.

     Curve Point

Arc dist, li, (ft)

Defl angle,δi

Radial chord, ci

EC      27+19.681 Bk 479.965 27°30'00.0" 461.748
       27+00 460.284 26°22'20.4" 444.203
       26+00 360.284 20°38'33.9" 352.540
       25+00 260.284 14°54'47.4" 257.355
       24+00 160.284 9°11'01.0" 159.599
       23+00 60.284 3°27'14.5" 60.248
BC      22+39.716 0.000 0°00'00.0" 0.000

 

Math checks:

Arc distances between successive full stations differ by 100.00 ft.

At the EC:

Arc distance should equal the curve length

Defl angle should equal Δ/2

Radial chord shoud equal Long Chord

The difference between deflection angles at successive full staions should be D/2.

These checks have been met.

Remember that we carried additonal decimal places to minimize rounding errors. Rounding can become quite pronounced for long flat curves so computation care must be exercised. Once the table has been computed, the final values can be shown to reasonable accuracy levels. For example, distances can be shown to 0.01 ft and deflection angles to 01".

d. Summary

Computing and staking curve points by the radial chord method is simple and straightforward. Using the curve equations, any point on the curve can be computed and staked, not just those included in the curve table.

However, in the field, it may not be the most efficient way to stake a curve using modern instrumentation, particularly with long curves which may have chords thousands of feet long at very small deflection angles. Most contemporay survey computations and fieldwork use coordinates giving greater stakeout flexibility. While the radial chord method may not be used for stake out, it is well adapted to coordinate computations as we'll see in the next section.