Middle Ordinate and External Distance

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Tangent Distance

Intermediate Engineering

Solving and laying out a simple curve

Middle Ordinate and External Distance

Two commonly used formulas for the middle ordinate (M) and the external distance (E) are as follows:

DEFLECTION ANGLES AND CHORDS

From the preceding discussions, one may think that laying out a curve is simply a matter of locating the center of a circle, where two known or computed radii intersect, and then swinging the arc of the circular curve with a tape. For some applications, that can be done; for example, when you are laying out the intersection and curbs of a private road or driveway with a residential street. In this case, the length of the radii you are working with is short. However, what if you are laying out a road with a 1,000- or 12,000- or even a 40,000-foot radius? Obviously, it would be impracticable to swing such radii with a tape. In usual practice, the stakeout of a long-radius curve involves a combination of turning deflection angles and measuring the length of chords (C, Cl, or CZ as appropriate). A transit is set up at the PC, a sight is taken along the tangent, and each point is located by turning deflection angles and measuring the chord distance between stations. This procedure is illustrated in figure 11-9. In this figure, you see a portion of a curve that starts at the PC and runs through points (stations) A, B, and C. To establish the location of point A on this curve, you should set up your instrument at the PC, turn the required deflection angle (all/2), and then measure the required chord distance from PC to point A. Then, to establish point B, you turn deflection angle D/2 and measure the required chord distance from A to B. Point C is located similarly.

As you are aware, the actual distance along an arc is greater than the length of a corresponding chord; therefore, when using the arc definition, either a correction is applied for the difference between arc

Figure 11-9.-Deflection angles and chords.

length and chord length, or shorter chords are used to make the error resulting from the difference negligible. In the latter case, the following chord lengths are commonly used for the degrees of curve shown:

100 feet0 to 3 degrees of curve

50 feet3 to 8 degrees of curve

25 feet8 to 16 degrees of curve

10 feet-over 16 degrees of curve

The above chord lengths are the maximum distances in which the discrepancy between the arc length and chord length will fall within the allowable error for taping. The allowable error is 0.02 foot per 100 feet on most construction surveys; however, based on terrain conditions or other factors, the design or project engineer may determine that chord lengths other than those recommended above should be used for curve stakeout.

The following formulas relate to deflection angles: (To simplify the formulas and further discussions of deflection angles, the deflection angle is designated simply as d rather than d/2.)

Where:

d = Deflection angle (expressed in degrees)

C = Chord length

D = Degree of curve

d = 0.3 CD

Where:

d = Deflection angle (expressed in minutes)

C = Chord length

D = Degree of curve

Where:

d = Deflection angle (expressed in degrees)

C = Chord length

R = Radius.

Figure 11-1O.Laying out a simple curve.