Handbook of Civil Engineering Calculations

Applying Eq. 38a to find the orientation angle and using data from the previous calcu-
lation procedure, we find Os = 10.5°, 05 /(3« 5
2
) = 10.5°/[3(10
2
)] = 0.035° = 2.1'; nb = O; np =
3;^ = 6(3)(2.1') = 0
0
37.8
/
.

2. Find the deflection angles from the tangent through point 3
   Thus, by Eq. 39a: point 4, 8 = (6 + 4)(2.1') = 0°21'; point 5, 6 - (6 + 5)(2)(2.1') -
   0°46.2'; point 6, 5 = (6 + 6)(3)(2.1') = 1°15.6'; point 7, S = (6 + 7)(4)(2.1') = 1°49.2'.


3. Consider that the transit is set up at point 7 and a backsight is
   taken to point 3; compute the orientation angle

Thus nb = 3; np = 7; (^8) b = (14 + 3)(4)(2.1') - 2°22.8'.

4. Compute the deflection angles from the tangent through
   point 7
   Thus point 8, 8 = (14 + 8)(2.1') - 0°46.2'; point 9, 8 = (14 + 9)(2)(2.1') - 1°36.6' Sc, 8 =
   (14+10)(3)(2.1') = 2°31.2'.


5. Test the results obtained
   In Fig. 18, extend chord PF to its intersection with the main tangent, and let a denote the
   angle between these lines. Then

n

« = («/ +»/»p + HJ)-T7 (40)

jns

This result should equal the sum of the angles applied in staking the curve from the TS to
F. This procedure will be shown with respect to point 9.
For point 9, let P and F refer to points 7 and 9, respectively. Then a = (9^2 + 9 x
7 + 72 )(2.1') - 6°45.3'. Summing the angles leading from the TS to point 9, we get

Deflection angle from main tangent to point 3 0° 18.9'

Orientation angle at point 3 0°37.8'

Deflection angle from local tangent to point 7 1°49.2'

Orientation angle at point 7 2°22.8'

Deflection angle from local tangent to point 9 1°36.6'

Total 6°45.3'

This test may be applied to each deflection angle beyond point 3.

PLOTTING A PARABOLIC ARC

A grade of-4.6 percent is followed by a grade of+1.8 percent, the grades intersecting at
station 54 + 20 of elevation 296.30 ft (90.312 m). The change in grade is restricted to 2
percent in 100 ft (30.5 m). Compute the elevation of every 50-ft (15.24-m) station on the
parabolic curve, and locate the sag (lowest point of the curve). Apply both the average-
grade method and the tangent-offset method.

Calculation Procedure:

1. Compute the required length of curve
   Using the notation in Figs. 19 and 20, we have Ga = -4.6 percent; Gb = +1.8 percent;