Flood Control for the FE Civil Exam: Return Period, Spillways and Reservoir Routing

NCEES FE Civil spec: Water Resources and Environmental Engineering, E. Flood control (e.g., dams, routing, spillways). FE Reference Handbook: Civil Engineering chapter, Hydrology/Water Resources section (weir formulas); Engineering Probability and Statistics chapter.

What the FE Civil Spec Covers

Section E is flood control: dams, routing and spillways. Expect questions on the probability of a design flood, the discharge or head over a spillway crest, and how a reservoir attenuates an inflow hydrograph. These are usually short calculations once you pick the right relationship.

Key Concepts

Return period T: the average interval between events of a given size. Annual exceedance probability is 1/T.
Risk over a design life: the probability of at least one exceedance in n years, assuming independent years.
Spillways: a principal spillway passes frequent floods, an emergency spillway passes rare floods so the dam is never overtopped. Ogee crests, broad-crested and sharp-crested weirs all follow $Q \propto L H^{3/2}$ with different coefficients.
Routing: storage in a reservoir lowers and delays the outflow peak. For an uncontrolled spillway, the outflow peak occurs where the outflow hydrograph crosses the falling limb of the inflow.

Core Formulas

p = \frac{1}{T} Risk = 1 - (1 - \frac{1}{T})^{n}

Rectangular weir (C about 3.33 US or 1.84 SI for a sharp crest, suppressed):

Q = C L H^{3/2}

Contracted rectangular weir with two end contractions, and a 90 degree V-notch:

Q = C (L - 0.2 H) H^{3/2} Q = C H^{5/2}

Storage indication (modified Puls) routing over a time step:

\frac{2 S _{2}}{Δ t} + O_{2} = (I_{1} + I_{2}) + (\frac{2 S _{1}}{Δ t} - O_{1})

Where It Is in the FE Reference Handbook

Weir formulas with US and SI coefficients are in the Civil Engineering chapter, Hydrology/Water Resources section (search "Weir"). The handbook does not give a routing equation, so know the storage equation and the storage indication form. For return period and binomial risk, the Engineering Probability and Statistics chapter has the binomial distribution.

Worked Example

An emergency spillway is a 15 m long rectangular weir with C = 1.84 (SI). (a) Find the discharge when the head over the crest is 1.2 m. (b) The spillway is sized for the 100-year flood. What is the chance that this flood is equaled or exceeded at least once during a 30-year project life?

Step 1, spillway discharge:

Q = 1.84 (15) (1.2)^{3/2} = 1.84 (15) (1.315) = 36.3 m^{3} / s

Step 2, chance of no exceedance in 30 years:

(1 - 0.01)^{30} = 0.9 9^{30} = 0.740

Step 3, risk:

Risk = 1 - 0.740 = 0.260 \approx 26%

Answer: about 36 m³/s over the crest, and about a 26% chance of at least one 100-year flood in 30 years.

How FE Questions Are Framed

Return period questions are often written in words: the chance that a structure designed for the 50-year flood is exceeded at least once in 20 years, or the return period that gives a 10% risk over a 25-year life. For the second form, solve the risk equation for T. Using the numbers in that example, $1 - 0.10 = (1 - 1/ T)^{25}$ gives T of about 238 years.

Spillway questions either give the head and ask for discharge, or give the design discharge and ask for the head (so you can check freeboard to the dam crest) or the crest length. Routing questions give a storage and outflow table and an inflow hydrograph, and ask for storage or outflow at the end of one time step. Compute the right side of the routing equation first, then look up the storage and outflow pair that matches the left side, interpolating between rows of the table as needed.

Common Mistakes

Computing risk as n/T (30/100 = 30%). That overstates the risk and exceeds 100% for long lives.
Using the SI weir coefficient with feet, or 3.33 with meters.
Treating Q as linear in H. Head is raised to the 3/2 power, so solving for H needs the 2/3 power.
Dropping the 2S/Δt terms or the sign on O₁ in the routing equation.
Forgetting end contractions when the weir is narrower than the approach channel.

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

An emergency spillway discharges over a rectangular sharp-crested weir with a crest length of 6 m and a weir coefficient of 1.84 (SI units). For a design discharge of 15 m³/s, the head over the crest is most nearly: See the worked solution or practice the Fluids & Hydraulics deck
A rectangular sharp-crested weir with a crest length of 1.0 m is installed in a channel that is wider than the weir, so the nappe contracts at both ends. The head over the crest is 0.6 m. Using C = 1.84 (SI units), the discharge is most nearly: See the worked solution or practice the Hydrology & Stormwater deck
A rectangular sharp-crested weir has b = 1.2 m, H = 0.40 m and Cd = 0.62. Using Q = (2/3)·Cd·b·√(2g)·H^(3/2), Q is most nearly: See the worked solution or practice the Fluids & Hydraulics deck
A flood threshold has an annual exceedance probability of 1/50 (0.02). Years are independent and the probability is stationary. What is the probability of at least one exceedance during a 30-year period? See the worked solution or practice the Hydrology & Stormwater deck
Modified Puls routing uses Δt = 600 s and 2S₂/Δt + O₂ = 2S₁/Δt − O₁ + I₁ + I₂. At the start of the step S₁ = 300 m³, O₁ = 0.5 m³/s, I₁ = 0.5 m³/s and I₂ = 1.5 m³/s. The storage-outflow rating pairs (S in m³, O in m³/s) are (300, 0.50), (450, 0.75), (600, 1.00), (750, 1.25); interpolate linearly. Find S₂. See the worked solution or practice the Hydrology & Stormwater deck

Related Calculators and Examples

Frequently Asked Questions

Does a 100-year flood happen once every 100 years?

No. It has a 1% chance of being equaled or exceeded in any single year. Over a 30-year period the chance of at least one such flood is about 26%, and two 100-year floods can occur in consecutive years.

What weir coefficient should I use for a rectangular weir?

Use the value the problem gives. If none is given, a common sharp-crested value is about 3.33 in US units (feet and cfs) or 1.84 in SI units (meters and cubic meters per second) for Q = C L H to the 1.5 power.

What is reservoir routing?

It is the step-by-step solution of the storage equation, inflow minus outflow equals the rate of change of storage, to find how a reservoir or basin reduces and delays a flood peak. The storage indication (modified Puls) method is the standard hand method.

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