Calculate flow or head for all eight weir shapes in the FHWA Hydraulic Toolbox, including paved or gravel irregular roadway crests with segment-by-segment tailwater submergence correction.
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Choose one of the eight FHWA weir shapes and enter either head or flow.
For educational purposes only. Not a substitute for professional engineering judgment.
FHWA weir equations
Rectangular and Cipolletti weirs use the standard three-halves-power relationship from FHWA HDS-4:
Q = CD L H3/2
The preset V-notch choices use the corresponding five-halves-power relationship, Q = CVH5/2. The coefficients match those embedded in FHWA Hydraulic Toolbox: 3.100 for rectangular, 3.367 for Cipolletti, and 2.500, 1.443, 1.035, and 0.497 for the 90-, 60-, 45-, and 22.5-degree V-notches in US customary units.
Coefficients are unit-bearing. Switching to metric applies the same exact conversion used by the FHWA executable, CSI = CUS√0.3048.
Irregular roadway overtopping
For an irregular crest, head is measured above the lowest entered crest elevation. The calculator clips each station/elevation segment at the water surface and sums:
Q = Σ CD,i Li Hi3/2
The segment coefficient is CD,i = Cr,ikt,i. The roadway coefficient Cr depends on surface, local upstream head, and crest width; the submergence factor kt depends on the local tailwater-to-head ratio. Both are linearly interpolated from FHWA HDS-5 Figure 3.11 using the same ordinates as the desktop package.
FHWA verification example
For a 4 ft rectangular weir with 1.2 ft of head and CD = 3.1, the calculator gives 16.300223 cfs. FHWA Hydraulic Toolbox displays the same result rounded to 16.300 cfs.
Frequently asked questions
Which weir shapes does this calculator support?
It implements all eight FHWA Hydraulic Toolbox choices: rectangular, Cipolletti, 90-, 60-, 45-, and 22.5-degree V-notches, a user-defined V-notch coefficient, and an irregular station/elevation roadway crest.
Can the calculator solve for either flow or head?
Yes. Enter head to calculate discharge, or enter discharge to solve the rating equation for head. Regular shapes have direct inverse equations; irregular crests use a converged numerical solution.
How is an irregular roadway crest calculated?
The crest is defined with station/elevation points. Each line segment is clipped at the upstream water surface, evaluated at its midpoint head, and assigned a paved, gravel, or user-entered coefficient. Segment flows are then summed in accordance with the FHWA HDS-5 roadway-overtopping procedure.
How does tailwater affect an irregular weir?
For each wet segment, the downstream-to-upstream head ratio selects an HDS-5 submergence factor. That factor multiplies the segment roadway coefficient before its discharge is added to the total. FHWA does not expose tailwater correction in the regular-shape modes.
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Last verified: August 2026