1 r0 v2
∆Z =
∆r
g ri r
Total superelevation is 0.040 meters.
Based on relationships, V = Q/A = 1.10 m/s, the following are obtained:
Woodward's Equation 2.160
(1.10)
2
2
∆Z = V (r0 - ri) =
(156.4 - 106.7) = 0.047 m
9.81(131.4)
g rc
Ippen and Drinker's Equation 2.162
2
1.10 2
2W
2 x 49.7
1
1
∆Z = V
= 0.042 m
=
x
2
2
2g rc W
2 x 9.81
131.4
49.7
1 - 2r
1 -
2 x 131.4
c
Ippen and Drinker's Equation 2.163
2
2
= 1.10 x 2 x 49.7
2W
1
1
V
= 0.041 m
∆Z =
2g rc
2
4.97 2
W 2 x 9.81
131.4
1 -
1 -
2
12 x 131.4 2
12 rc
Combined Free and Forced Vortex Equation 2.165
Vmax = 1.49 m/s
(
)
rc 1.49 2 106.7 131.4
2
2
2
2
2
ri
V max
- =
2-
= 0.0718 m
r 2 x 9.81 2 - 131.4 - 156.4
∆Z =
r
2g
c
o
The equations give comparable results (0.040 to 0.0718 m). Equation 2.158 which integrates
across the section using the velocity distribution is the most exact. But using the value 0.072 m
provides a safety factor.
2.14.4 PROBLEM 4 Maximum Stream Constriction Without Causing Backwater
(Neglecting Energy Losses)
A stream is rectangular in shape and 30.48 m wide. The design discharge is 141.6 cms and the
uniform depth for this discharge is 3.05 m. Neglecting energy losses what is the maximum
amount of constriction that a bridge can impose without causing backwater.
2.75
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