Math on diversion & head work

1. Determine the percentage pressures at various key points in figure below. Also determine the exit
gradient and plot the hydraulic gradient line for pond level on upstream and no flow on downstream

2. U/S pile
57m
6m
b =57m
d=6m
=b/d=9.5
=
1+ 1+𝛼2
2
=5.28
𝐸=100%
𝐷 =
100
𝜋
𝑐𝑜𝑠−1 1−𝜆
𝜆
=80.12
𝐶 =
100
𝜋
𝑐𝑜𝑠−1 2−𝜆
𝜆
=71.26
E
D
C

3. D/S pile
57m
10.3m
b =57m
d=10.3m
=b/d=5.53
=
1+ 1+𝛼2
2
=3.3
𝐸=
100
𝜋
𝑐𝑜𝑠−1 𝜆−2
𝜆
=37
𝐷 =
100
𝜋
𝑐𝑜𝑠−1 𝜆−1
𝜆
=25.4
𝐶 =0
E
D
C

4. Intermediate pile
57m
6m
b1 =15.8m b2=40m
d=6m
1=b1/d=2.63
2=b2/d=6.67
𝜆 =
1+𝛼1
2+ 1+𝛼2
2
2
=4.78
1=
1+𝛼1
2− 1+𝛼2
2
2
=-1.96
𝐸=
100
𝜋
𝑐𝑜𝑠−1 𝜆1−1
𝜆
=71.32
𝜙𝐷 =
100
𝜋
𝑐𝑜𝑠−1 𝜆1
𝜆
=63.5
𝐶 =
100
𝜋
𝑐𝑜𝑠−1 𝜆1+1
𝜆
=56.5
E
D
C

5. 71.32%
63.5%
56.5%
100%
0
80.12%
71.26% 37%
25.4%

6. Corrections for Pile 1

7. 𝑪𝟏 =71.26+1.88+1.5=74.64

8. Corrections for Pile 2

9. 𝑬𝟐= 71.32-1.88-1.17 = 68.27

11. 𝑪𝟐 =56.5+2.89+1.17-0.34=60.22

12. Corrections for Pile 3

13. 𝑬𝟑 =37-1.02-1.46=34.52

16. 71.32-1.88-1.17
= 68.27
63.5%
56.5+2.89+1.17-0.34
=60.22
100%
0
80.12%
71.26+1.88+1.5=
74.64
37-1.02-1.46
=34.52
25.4%