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 40,000foot radius? Obviously, it would be impracticable to
swing such radii with a tape. In usual practice, the stakeout of a longradius
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
119. 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 119.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 feet—0 to 3 degrees of curve
50 feet—3 to 8 degrees of curve
25 feet—8 to 16 degrees of curve
10 feetover 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 111O.—Laying out a simple curve.
