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Gumbel design flood for highway bridge, 4% risk

Short answer

Design return period from 4% risk in 50 years: T=1225 years. Using Gumbel linearity in reduced variate with known floods (150 m³/s at 50 yr, 650 m³/s at 300 yr), design flood ≈ 1041 m³/s.

A highway bridge has to be designed with an expected life of 50 years and an allowable flood risk of 4%. The flood data of bridge site were well fitted to Gumbel EV distribution and discharges for 50 and 300 years return period are found to be 150 and 650 m3^3/sec respectively. Estimate the frequency and magnitude of design flood for the bridge.

Frequency (return period) of the design flood

Risk R=0.04R=0.04 in a life n=50n=50 years:

R=1−(1−1T)n ⇒ 1T=1−(0.96)1/50 ⇒ T=1225.3 yearsR=1-\left(1-\frac1T\right)^{n}\ \Rightarrow\ \frac1T=1-(0.96)^{1/50}\ \Rightarrow\ T=1225.3 \text{ years}

The design flood is the 1225-year flood, with annual exceedance probability 1/T=0.000821/T=0.00082.

Magnitude of the design flood

Gumbel's equation is linear in the reduced variate yT=−ln⁡[ln⁡(T/(T−1))]y_T=-\ln[\ln(T/(T-1))], so the two known floods define the line:

T (yr)yTy_TxTx_T (m³/s)
503.9019150
3005.7021650
12257.1106?
σSn=650−1505.7021−3.9019=277.75 m3/s\frac{\sigma}{S_n}=\frac{650-150}{5.7021-3.9019}=277.75 \text{ m}^3/\text{s} xT=150+277.75 (7.1106−3.9019)=1041 m3/sx_T=150+277.75\,(7.1106-3.9019)=1041 \text{ m}^3/\text{s}

Answer: design return period ≈ 1225 years; design flood ≈ 1041 m³/s.

✓ Answer checked by Claude Sonnet 5.5

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