### Section 5. Chord Definition

#### a. General

Recall from Section 2 there are two definitions for degree of curvature, Figure C-32.

a. Arc Definition |
b. Chord Definition |

Figure C-32 |

Arc definition is the central angle for a 100.00 ft arc; chord is for a 100.00 ft chord.

Since the geometry is slightly different, the mathematical relationship between R and D_{c} is Equation C-25.

Equation C-25 |

Equations C-3 through C-6 for T, LC, E, and M can be used with no changes.

Equation C-2 for L does not yield the total arc length of the curve from the BC to EC. It returns the sum of the subchords, Figure C-33 and Equation C-26.

Figure C-33 Sum of the Chords |

Equation C-26 |

Where:

c_{f} |
first partial chord (<100.00 ft) |

n | number of full 100.00 ft chords on the curve |

c_{l} |
last partial chord (<100.00 ft) |

#### b. Stationing

The same equations are used to compute endpoint stationing for arc and chord definition curves.

Equation C-27 | |

Equation C-28 | |

Equation C-29 |

A notable difference from the arc definition is that curve stationing is along the *chords*, not the *arcs*.

#### c. Radial Chord Deflection Method

Because stationing is through the chords, deflection angles to curve points are can be computed by adding incremental deflection angles. The radial chord to each curve point is determined using Equation C-15.

*First partial chord, c*_{f}

c_{f}is the difference between the first full curve station and the BC station.

The central angle and incremental deflection angles are:

Equation C-30 | |

Equation C-31 |

*Last partial chord, c*_{l}*:*

c_{l}is the difference between the last full curve station and the EC Back station.

The central angle and incremental deflection angles are:

Equation C-32 | |

Equation C-33 |

*Nominal chord*

The chord between adjacent full stations is 100.00 ft.

The incremental deflection angle is:

Equation C-34 |

Starting with the first incremental deflection angle, the procedure to compute total deflection angle to curve points is:

#### d. Example Problem

PI Station is 59+45.00, Δ angle is 30°00'00", a 7°00'00" degree of curvature (chord def) will be used.

Compute the curve table.

(1) Curve components

##### (2) Stationing

##### (3) Incremental deflection angles

first partial chord

last partial chord

nominal chord

##### (4) Curve Table

Set up table with stations and incremental deflection angles

Sta | Inc Defl | Total Defl | Radial Chord | |

EC Bk | 61+54.115 | 1°53'39" | ||

61+00 | 3°30'00" | |||

60+00 | 3°30'00" | |||

59+00 | 3°30'00" | |||

58+00 | 2°36'21" | |||

BC | 57+25.544 | 0°00'00 |

Compute total deflection angle by adding incremental deflection angles

Sta | Inc Defl | Total Defl | Radial Chord | |

EC Bk | 62+65.715 | 1°53'39" | 15°00'00" | |

62+00 | 3°30'00" | 13°06'21" | ||

61+00 | 3°30'00" | 9°36'21" | ||

60+00 | 3°30'00" | 6°06'21" | ||

59+00 | 2°36'21" | 2°36'21" | ||

BC | 58+37.174 | 0°00'00 | 0°00'00" |

Total deflection angle to EC is 30°00'00"/2 = 15°00'00"check

Compute radial chord to curve point using:

Sta | Inc Defl | Total Defl | Radial Chord | |

EC Bk | 62+65.715 | 1°53'39" | 15°00'00" | 423.96 |

62+00 | 3°30'00" | 13°06'21" | 371.43 | |

61+00 | 3°30'00" | 9°36'21" | 273.34 | |

60+00 | 3°30'00" | 6°06'21" | 174.42 | |

59+00 | 2°36'21" | 2°36'21" | 74.47 | |

BC | 58+37.174 | 0°00'00 | 0°00'00" | 0.00 |

Radial chord to EC is 423.96' = LCcheck

#### e. Arc Length

Should it be needed, the *arc* length, L_{A}, from the BC to the EC, can be computed using Equation C-35.

Equation C-35 |

Δ_{r} is the Δ angle in radians, Equation C-36.

Equation C-36 |