sm_ch14

Water Resources Engineering

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14 – 1 Chapter 14 14.4.1. The solution is given in the following table. The values given in column (2) are computed from the values given in column (1). While the values in column (3) are given values, those in column (4) are computed using the formula ±² ±² ± ² >@ ± ² j 1 j T T 1 j j Q Q P Q Q P T D T D 2 1 D E d ³ d ´ ' ´ ´ The first value in column (5) is the sum of all the incremental damage costs in column (4). The remaining values in this column are determined by subtracting the incremental damage cost of the corresponding year from the total value for the previous return period. The total cost in column (7) is the sum of the damage risk cost and the capital cost. The data given in the table is also illustrated graphically as shown. The optimal return period for this example is 25 years which corresponds to the lowest total annual cost of $80,500. (1) Return period T (years) (2) Annual excedence probability (3) Damage D(T) ($) (4) Incremental expected damage ($/year) (5) Damage risk cost ($/year) (6) Capital cost ($/year) (7) Total cost ($/year) 1 2 5 10 15 20 25 50 100 200 1.000 0.500 0.200 0.100 0.067 0.050 0.040 0.020 0.010 0.005 0 40000 120000 280000 354000 426000 500000 600000 800000 1000000 - 10000 24000 20000 10566 6500 4630 11000 7000 4500 98196 88196 64196 44196 33630 27130 22500 11500 4500 0 0 6000 28000 46000 50000 54000 58000 80000 120000 160000 98196 94196 92196 90196 83630 81130 80500 91500 124500 160000 Cost versus return period 0 20000 40000 60000 80000 100000 120000 140000 160000 180000 0 50 100 150 200 Return period (years) Cost ($/year) Damage Capital Total
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14 – 2 14.4.2. The discharge-frequency curve, the rating curve and the storage-damage curve are shown below, in the same order. 1000 10000 100000 0.01 0.1 1 10 100 Excedence probability (%) Discharge (cfs) Frequency curve 20 25 30 35 40 45 50 10000 15000 20000 25000 30000 Discharge (cfs) Stage (ft) Rating curve 25 30 35 40 45 50 0 1 02 03 04 05 06 0 Damage ($1,000,000) Stage-damage curve 14.4.3. Since the dike becomes completely ineffective when over-topped, the damage will be the same as the existing condition when the discharge is greater than 15000 cfs and zero when Q d 15000 cfs. At the excedence probability of 7%, the damage drops to zero since Q = 15000 cfs at this probability. The corresponding Figure is shown below.
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14 – 3 Excedence probability (%) Damage ($10 6 ) 20 0 10 0 7 0 7 10 5 13 2 22 1 30 0.5 40 0.2 50 0.1 54 0.05 57 0 10 20 30 40 50 60 0.01 0.1 1 10 100 Excedence probability (%) Damage (%) Damage-excedence probability curve 14.4.4. For a protection of up to 15000 cfs, the capacity of the upstream diversion is 15000 – 12000 = 3000 cfs. In this scenario, inflows in excess of 12000 cfs will be diverted upstream up to a maximum of 3000 cfs. The following table gives the computation of the damage frequency curve for the this scenario, and the subsequent Figure shows the resulting curve.
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sm_ch14 - 14 1 Chapter 14 14.4.1. The solution is given in...

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