What are the HDS-5 inlet control coefficients K, M, c and Y?

They are regression constants FHWA fitted to the inlet-control nomographs so headwater can be computed by equation. K and M describe the unsubmerged (weir-like) range: HW/D = Hc/D + K[Q/(A D^0.5)]^M − 0.5S (Form 1) or HW/D = K[Q/(A D^0.5)]^M (Form 2). c and Y describe the submerged (orifice-like) range: HW/D = c[Q/(A D^0.5)]^2 + Y − 0.5S. Each culvert shape and inlet edge has its own set, tabulated in HDS-5 Table 9 (2nd edition) or Appendix A Table A.1 (3rd edition).

What are the inlet control constants for a square-edge concrete pipe with headwall?

For circular concrete pipe, square edge with headwall (HDS-5 Chart 1, Scale 1): K = 0.0098, M = 2.0 (Form 1), c = 0.0398, Y = 0.67. Groove end with headwall is K = 0.0018, M = 2.0, c = 0.0292, Y = 0.74; groove end projecting is K = 0.0045, M = 2.0, c = 0.0317, Y = 0.69.

When is a culvert inlet unsubmerged versus submerged in HDS-5?

HDS-5 applies the unsubmerged equations up to about Q/(A D^0.5) = 3.5 in English units (1.93 SI) and the submerged equation above about 4.0 (2.21 SI). Between those values neither equation is strictly valid and the headwater is taken by interpolating the two results, which is what the transition zone on the nomographs represents.

How do I use the HDS-5 inlet control equations in SI units?

Multiply the discharge term by Ku = 1.811 before raising it to the power: use Ku·Q/(A D^0.5) with Q in m³/s, A in m² and D in m. In English units Ku = 1.0 with Q in cfs, A in ft² and D in ft. The K, M, c and Y constants are the same in both systems. A 900 mm concrete pipe, square edge with headwall, at 1.5 m³/s and 1% slope gives Ku·Q/(A D^0.5) = 4.50 and HW/D = 1.47, so HW = 1.32 m.

Why does a mitered inlet use +0.7S instead of −0.5S?

The slope term corrects for the fall between the inlet control section and the point where headwater is measured. For most inlets the control section sits slightly downstream of the face, so headwater is reduced by about 0.5S·D. For inlets mitered to the embankment slope the geometry is reversed and HDS-5 specifies +0.7S in place of −0.5S, which raises the computed headwater.

Is the entrance loss coefficient Ke the same thing as these constants?

No. Ke is used only in outlet-control calculations, as the entrance term in H = (1 + Ke + 29n²L/R^1.33)·V²/2g. The K, M, c, Y constants replace the inlet-control nomographs and already contain the inlet's hydraulic efficiency. A design uses both: compute inlet-control headwater from K, M, c, Y and outlet-control headwater with Ke, then take the larger.

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HDS-5 Inlet Control Coefficients (K, M, c, Y)

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 8, 2026

The regression constants FHWA fitted to the inlet-control nomographs, so culvert headwater can be computed by equation instead of read from a chart. Every row of HDS-5 Table 9 (2nd edition) / Appendix A Table A.1 (3rd edition) is here: shape, inlet edge, equation form, K, M for the unsubmerged range and c, Y for the submerged range. Constants are identical in SI and English units; only the K_u factor on the discharge term changes.

The Three Equations

Unsubmerged, Form 1 (weir-like; used with charts that list Form 1)

HW_i/D = H_c/D + K·[K_u·Q/(A·D^0.5)]^M − 0.5·S

Unsubmerged, Form 2 (simpler fit; used with charts that list Form 2)

HW_i/D = K·[K_u·Q/(A·D^0.5)]^M

Submerged (orifice-like; all charts)

HW_i/D = c·[K_u·Q/(A·D^0.5)]² + Y − 0.5·S

HW_i = headwater depth above the inlet-control section invert; D = interior barrel height; H_c = specific head at critical depth, d_c + V_c²/2g; Q = discharge; A = full barrel area; S = barrel slope. K_u = 1.0 English (cfs, ft², ft) or 1.811 SI (m³/s, m², m).
Applicability: unsubmerged forms up to about Q/(A·D^0.5) = 3.5 English (1.93 SI); submerged above about 4.0 English (2.21 SI). Between them, interpolate.
Mitered inlets: use +0.7·S in place of −0.5·S as the slope correction.

Circular Culverts

Chart	Shape / material	Scale	Inlet edge description	Form	K	M	c	Y
1	Circular concrete	1	Square edge w/ headwall	1	0.0098	2.0	0.0398	0.67
1	Circular concrete	2	Groove end w/ headwall	1	0.0018	2.0	0.0292	0.74
1	Circular concrete	3	Groove end projecting	1	0.0045	2.0	0.0317	0.69
2	Circular CMP	1	Headwall	1	0.0078	2.0	0.0379	0.69
2	Circular CMP	2	Mitered to slope (use +0.7S)	1	0.0210	1.33	0.0463	0.75
2	Circular CMP	3	Projecting	1	0.0340	1.50	0.0553	0.54
3	Circular	A	Beveled ring, 45° bevels	1	0.0018	2.50	0.0300	0.74
3	Circular	B	Beveled ring, 33.7° bevels	1	0.0018	2.50	0.0243	0.83
55	Circular, tapered inlet	1	Smooth tapered inlet throat	2	0.534	0.555	0.0196	0.90
55	Circular, tapered inlet	2	Rough tapered inlet throat	2	0.519	0.64	0.0210	0.90

Sizing the crossing inside a real watershed, not one barrel? HydroComplete runs inlet and outlet control on every culvert in the model against the routed hydrograph and reports the governing headwater.

Rectangular Box Culverts

Chart	Shape / material	Scale	Inlet edge description	Form	K	M	c	Y
8	Rectangular box	1	30° to 75° wingwall flares	1	0.026	1.0	0.0347	0.81
8	Rectangular box	2	90° and 15° wingwall flares	1	0.061	0.75	0.0400	0.80
8	Rectangular box	3	0° wingwall flares (parallel extensions)	1	0.061	0.75	0.0423	0.82
9	Rectangular box	1	45° wingwall flare, d = 0.043D	2	0.510	0.667	0.0309	0.80
9	Rectangular box	2	18° to 33.7° wingwall flare, d = 0.083D	2	0.486	0.667	0.0249	0.83
10	Rectangular box	1	90° headwall w/ 3/4-in chamfers	2	0.515	0.667	0.0375	0.79
10	Rectangular box	2	90° headwall w/ 45° bevels	2	0.495	0.667	0.0314	0.82
10	Rectangular box	3	90° headwall w/ 33.7° bevels	2	0.486	0.667	0.0252	0.865
11	Rectangular box	1	3/4-in chamfers; 45° skewed headwall	2	0.545	0.667	0.04505	0.73
11	Rectangular box	2	3/4-in chamfers; 30° skewed headwall	2	0.533	0.667	0.0425	0.705
11	Rectangular box	3	3/4-in chamfers; 15° skewed headwall	2	0.522	0.667	0.0402	0.68
11	Rectangular box	4	45° bevels; 10°–45° skewed headwall	2	0.498	0.667	0.0327	0.75
12	Rectangular box, 3/4-in chamfers	1	45° non-offset wingwall flares	2	0.497	0.667	0.0339	0.803
12	Rectangular box, 3/4-in chamfers	2	18.4° non-offset wingwall flares	2	0.493	0.667	0.0361	0.806
12	Rectangular box, 3/4-in chamfers	3	18.4° non-offset wingwall flares, 30° skewed barrel	2	0.495	0.667	0.0386	0.71
13	Rectangular box, top bevels	1	45° wingwall flares, offset	2	0.497	0.667	0.0302	0.835
13	Rectangular box, top bevels	2	33.7° wingwall flares, offset	2	0.495	0.667	0.0252	0.881
13	Rectangular box, top bevels	3	18.4° wingwall flares, offset	2	0.493	0.667	0.0227	0.887
16–19	Corrugated metal box	2	90° headwall	1	0.0083	2.0	0.0379	0.69
16–19	Corrugated metal box	3	Thick wall projecting	1	0.0145	1.75	0.0419	0.64
16–19	Corrugated metal box	5	Thin wall projecting	1	0.0340	1.5	0.0496	0.57
57	Rectangular, tapered inlet	1	Tapered inlet throat	2	0.475	0.667	0.0179	0.97
58	Rectangular concrete	1	Side-tapered, less favorable edges	2	0.56	0.667	0.0446	0.85
58	Rectangular concrete	2	Side-tapered, more favorable edges	2	0.56	0.667	0.0378	0.87
59	Rectangular concrete	1	Slope-tapered, less favorable edges	2	0.50	0.667	0.0446	0.65
59	Rectangular concrete	2	Slope-tapered, more favorable edges	2	0.50	0.667	0.0378	0.71

Box vs. non-box constants are not interchangeable. HDS-5 is explicit that rectangular constants must not be used for circular, arch or pipe-arch shapes and vice versa. For a new shape without a chart, pick the tabulated shape closest in geometry and inlet edge and generate curves from its constants.

Ellipse, Pipe-Arch and Arch Culverts

Chart	Shape / material	Scale	Inlet edge description	Form	K	M	c	Y
29	Horizontal ellipse, concrete	1	Square edge w/ headwall	1	0.0100	2.0	0.0398	0.67
29	Horizontal ellipse, concrete	2	Groove end w/ headwall	1	0.0018	2.5	0.0292	0.74
29	Horizontal ellipse, concrete	3	Groove end projecting	1	0.0045	2.0	0.0317	0.69
30	Vertical ellipse, concrete	1	Square edge w/ headwall	1	0.0100	2.0	0.0398	0.67
30	Vertical ellipse, concrete	2	Groove end w/ headwall	1	0.0018	2.5	0.0292	0.74
30	Vertical ellipse, concrete	3	Groove end projecting	1	0.0095	2.0	0.0317	0.69
34	Pipe-arch, 18-in corner radius, CM	1	90° headwall	1	0.0083	2.0	0.0379	0.69
34	Pipe-arch, 18-in corner radius, CM	2	Mitered to slope (use +0.7S)	1	0.0300	1.0	0.0463	0.75
34	Pipe-arch, 18-in corner radius, CM	3	Projecting	1	0.0340	1.5	0.0496	0.57
35	Pipe-arch, 18-in corner radius, CM	1	Projecting	1	0.0300	1.5	0.0496	0.57
35	Pipe-arch, 18-in corner radius, CM	2	No bevels	1	0.0088	2.0	0.0368	0.68
35	Pipe-arch, 18-in corner radius, CM	3	33.7° bevels	1	0.0030	2.0	0.0269	0.77
36	Pipe-arch, 31-in corner radius, CM	1	Projecting	1	0.0300	1.5	0.0496	0.57
36	Pipe-arch, 31-in corner radius, CM	2	No bevels	1	0.0088	2.0	0.0368	0.68
36	Pipe-arch, 31-in corner radius, CM	3	33.7° bevels	1	0.0030	2.0	0.0269	0.77
41–43	Arch, corrugated metal	1	90° headwall	1	0.0083	2.0	0.0379	0.69
41–43	Arch, corrugated metal	2	Mitered to slope (use +0.7S)	1	0.0300	1.0	0.0463	0.75
41–43	Arch, corrugated metal	3	Thin wall projecting	1	0.0340	1.5	0.0496	0.57
56	Elliptical inlet face, tapered	1	Tapered inlet, beveled edges	2	0.536	0.622	0.0368	0.83
56	Elliptical inlet face, tapered	2	Tapered inlet, square edges	2	0.5035	0.719	0.0478	0.80
56	Elliptical inlet face, tapered	3	Tapered inlet, thin edge projecting	2	0.547	0.80	0.0598	0.75

Worked Examples

1. 36-in RCP, square edge with headwall, Q = 60 cfs, S = 0.01. Chart 1, Scale 1: c = 0.0398, Y = 0.67.

D = 3.0 ft, A = πD²/4 = 7.069 ft², A·D^0.5 = 12.24
Q/(A·D^0.5) = 60 / 12.24 = 4.90 > 4.0 → submerged equation
HW_i/D = 0.0398 × 4.90² + 0.67 − 0.5 × 0.01 = 0.956 + 0.665 = 1.621
HW_i = 1.621 × 3.0 = 4.86 ft

Same pipe and flow with a groove end w/ headwall (c = 0.0292, Y = 0.74) gives HW_i/D = 1.436, HW_i = 4.31 ft; a 33.7° beveled ring (c = 0.0243, Y = 0.83) gives 1.409 and 4.23 ft. The inlet edge alone is worth 0.6 ft of headwater here.

2. 4 ft × 4 ft concrete box, 45° wingwall flare, Q = 80 cfs. Chart 9, Scale 1 (Form 2): K = 0.510, M = 0.667.

A = 16 ft², A·D^0.5 = 32.0, Q/(A·D^0.5) = 2.50 < 3.5 → unsubmerged, Form 2 (no slope term, no H_c)
HW_i/D = 0.510 × 2.50^0.667 = 0.510 × 1.842 = 0.940
HW_i = 0.940 × 4.0 = 3.76 ft

At Q = 160 cfs the ratio is 5.0 and the submerged equation (c = 0.0309, Y = 0.80, S = 0.01) gives HW_i/D = 0.0309 × 25 + 0.80 − 0.005 = 1.568, HW_i = 6.27 ft.

3. SI check: 900 mm RCP, square edge with headwall, Q = 1.5 m³/s, S = 0.01.

A = 0.636 m², A·D^0.5 = 0.6035, Q/(A·D^0.5) = 2.485; × K_u 1.811 = 4.50 > 2.21 SI threshold → submerged
HW_i/D = 0.0398 × 4.50² + 0.67 − 0.005 = 1.471
HW_i = 1.471 × 0.9 = 1.32 m (the same pipe worked in English units, 2.953 ft and 52.97 cfs, gives 4.345 ft = 1.324 m)

Form 1 needs H_c. The Form 1 unsubmerged equation adds the specific head at critical depth, H_c/D = (d_c + V_c²/2g)/D, which for a circular pipe requires solving for d_c from Q²/g = A_c³/T_c. Form 2 charts absorb that term into K and M. Use the culvert calculator or the critical depth card for d_c. Design headwater is the larger of the inlet-control result here and the outlet-control result from the energy equation with K_e.

Source: FHWA, Hydraulic Design of Highway Culverts, Hydraulic Design Series No. 5, 2nd edition (Normann, Houghtalen & Johnston; FHWA-NHI-01-020, Sept. 2001, rev. May 2005), Table 8 (equations 26–28) and Table 9, pp. 192–194. The 3rd edition (FHWA-HIF-12-026, April 2012) carries the same constants as Appendix A, Table A.1. Chart and scale numbers refer to the HDS-5 nomographs. Values transcribed row by row from the published table; the six circular rows also match the constants in the pe-calc culvert calculator.

Ready to size a crossing? Open the culvert calculator → · Outlet control needs K_e and a barrel Manning's n.

Related cheat sheets and tools

The culvert hydraulics calculator applies the circular-pipe constants above and the outlet-control energy equation, and reports which control governs. Pair it with entrance loss coefficients K_e, Manning's n for culvert barrels, and critical depth for Form 1. Two step-by-step designs: driveway culvert, 25-year storm and 36-inch CMP road crossing. For a crossing inside a routed watershed model, see the 48-inch CMP inlet vs outlet control example from HydroComplete.

HDS-5 Inlet Control Coefficients (K, M, c, Y)

The Three Equations

Circular Culverts

Rectangular Box Culverts

Ellipse, Pipe-Arch and Arch Culverts

Worked Examples

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