Question

A bridge has an expected design life of 50 years. It is designed for a flood discharge of 1000 m\(^3\)/s, which corresponds to a return period of 100 years. Determine the risk (probability) that the design flood will be equalled or exceeded at least once during the design life of the bridge. (Enter the numerical value of risk in decimal form, correct up to three decimal places.)

Hint:

Remember the formula for hydrological risk: \(R = 1 - (1 - 1/T)^n\). It's a common application of binomial probability for "at least one success" over \(n\) trials.

Answer 1

Step 1: Understanding the Question:
The question asks for the 'risk' of a hydrological event. Risk is defined as the probability that an event of a certain magnitude (or greater) will occur at least once in a specified period (the design life).
Step 2: Key Formula or Approach:
The probability of an event with a return period \(T\) occurring in any given year is \(P = 1/T\).
The probability of the event not occurring in any given year is \(q = 1 - P\).
The probability of the event not occurring for \(n\) consecutive years is \(q^n = (1 - P)^n\).
The risk (\(R\)) is the probability of the event occurring at least once in \(n\) years, which is 1 minus the probability of it never occurring. \[ R = 1 - q^n = 1 - (1 - P)^n = 1 - \left(1 - \frac{1}{T}\right)^n \] Step 3: Detailed Explanation:
We are given the following values:

• Design life, \(n = 50\) years
• Return period, \(T = 100\) years

First, calculate the probability of the design flood being exceeded in a single year: \[ P = \frac{1}{T} = \frac{1}{100} = 0.01 \] Now, use the risk formula: \[ R = 1 - \left(1 - \frac{1}{100}\right)^{50} \] \[ R = 1 - (1 - 0.01)^{50} \] \[ R = 1 - (0.99)^{50} \] Calculating the value of \((0.99)^{50}\): \[ (0.99)^{50} \approx 0.605006 \] Now, calculate the risk: \[ R = 1 - 0.605006 = 0.394994 \] The question asks to correct the answer up to three decimal places. \[ R \approx 0.395 \] Step 4: Final Answer:
The risk that the design flood will be equalled or exceeded during the bridge's design life is 0.395.