Manning's n — Roughness Coefficient Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 15, 2026

Quick reference for open-channel and partial-flow conduit calculations. All values from Chow (1959) Table 5-6 unless noted. Print and pin above your desk.

Lined / Constructed Channels & Pipes

Material	n	Range
Smooth concrete, trowel-finished	0.013	0.011–0.015
Concrete, float-finished	0.015	0.013–0.016
Concrete, unfinished (against forms)	0.017	0.014–0.020
Gunite (shotcrete), good section	0.019	0.016–0.023
Concrete pipe (RCP), smooth wall	0.012	0.011–0.013
PVC pipe, smooth	0.011	0.010–0.013
HDPE, dual-wall (smooth interior)	0.012	0.010–0.013
HDPE, single-wall (corrugated interior)	0.024	0.020–0.027
Cast iron, coated	0.013	0.011–0.014
Corrugated metal pipe (2⅔ × ½ in)	0.024	0.022–0.027
Corrugated metal pipe (3 × 1 in)	0.027	0.025–0.030
Asphalt, smooth	0.013	0.012–0.015
Asphalt, rough	0.016	0.015–0.020
Riveted steel	0.016	0.013–0.017
Vitrified clay (sewer)	0.013	0.011–0.017
Brick, glazed	0.013	0.011–0.015
Brick, in cement mortar	0.015	0.012–0.018

Laboratory n vs design n — why published tables disagree. The values above are as-built, clean-conduit values. Many agencies require a higher design n for closed conduits to cover joint offsets, sediment, biofilm and long-term aging — commonly 0.013 for RCP and 0.012–0.013 for smooth-wall plastic, even though the clean-pipe values are 0.012 and 0.010–0.011. Neither number is wrong; they answer different questions. Check the governing state or municipal storm-sewer standard before designing, and say which basis you used.

The corrugated-HDPE trap. "HDPE pipe" is not one roughness. Dual-wall pipe with a smooth interior liner runs n ≈ 0.012; single-wall pipe with an exposed corrugated interior runs n ≈ 0.024 — twice the value, which roughly halves capacity at the same slope. Confirm the interior, not just the material.

Earth & Excavated Channels

Description	n	Range
Earth, clean, recently completed, straight	0.018	0.016–0.020
Earth, clean, after weathering, straight	0.022	0.018–0.025
Earth, gravelly, with some weeds, straight	0.025	0.022–0.030
Earth, weedy with stones	0.035	0.025–0.040
Earth, dense weeds, deep flow	0.080	0.050–0.120
Earth, rock cuts, smooth, uniform	0.035	0.025–0.040
Earth, rock cuts, jagged, irregular	0.040	0.035–0.050
Riprap, well-graded	0.040	0.030–0.045

Natural Streams (Bank-full, Top of Bank)

Description	n	Range
Clean, straight, full stage, no rifts or pools	0.030	0.025–0.033
Same as above, but more stones & weeds	0.035	0.030–0.040
Clean, winding, some pools and shoals	0.040	0.033–0.045
Same as above, with weedy & stony banks	0.045	0.035–0.050
Sluggish reach, weedy, deep pools	0.070	0.050–0.080
Mountain streams, gravel/cobbles, few boulders	0.040	0.030–0.050
Mountain streams, cobbles & boulders	0.050	0.040–0.070

Floodplain (Overbank Flow)

Description	n	Range
Pasture, short grass	0.030	0.025–0.035
Pasture, high grass	0.035	0.030–0.050
Cultivated, no crop	0.030	0.020–0.040
Cultivated, mature row crops	0.035	0.025–0.045
Light brush and trees, winter	0.050	0.035–0.060
Light brush and trees, summer	0.060	0.040–0.080
Heavy brush, summer	0.100	0.070–0.160
Trees, dense willows, straight	0.150	0.110–0.200
Trees, heavy stand, flooded with branches submerged	0.120	0.080–0.200

Sources: Chow, V.T. (1959). Open-Channel Hydraulics, Table 5-6. USGS Water-Supply Paper 2339 (Arcement & Schneider, 1989). USACE EM 1110-2-1601 for floodplain values.

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Hazen-Williams C — Pipe Roughness Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

For pressurized water-distribution head loss: h_f = 4.73·L·Q^1.852 / (C^1.852·D^4.87) (US units, h_f in ft, L & D in ft, Q in cfs). Use this for water at ~60°F in transmission and distribution mains. For other fluids, low-flow regimes, or where temperature matters, use Darcy-Weisbach instead.

Plastic Pipe

Material	C (new)	C (design)	Notes
PVC, smooth bore	150	140–150	AWWA C900/C905
HDPE, DR 11–17	150	140–150	AWWA C906; very stable with age
Polyethylene, small dia.	140	130–140	service lines

Iron & Steel Pipe

Material	C (new)	C (design / aged)	Notes
Ductile iron, cement-mortar lined	140	130–140	AWWA C151/C104; design value typically 130
Ductile iron, unlined (rare)	130	90–110	tuberculation reduces C dramatically
Cast iron, new	130	100–130
Cast iron, 20+ yr unlined	100	60–100	severe tuberculation possible
Steel, cement-mortar lined	140	130–140	AWWA C200/C205
Steel, coal-tar enamel lined	140	130–140	AWWA C203
Steel, riveted (legacy)	110	90–110
Galvanized steel	120	100–120

Concrete & Other

Material	C (new)	C (design / aged)	Notes
Concrete, smooth-formed	130	120–140
Concrete pressure pipe (PCCP)	140	130–140	AWWA C301/C303
Asbestos-cement (legacy)	140	130–140	still in service in older systems
Copper tubing	140	130–140	building service
Brass	130	120–130
Vitrified clay (sewer, full flow)	110	100–120

Design tip — pick the aged value, not the new. A 50-year water main analysis on the C-new value will significantly under-predict head loss. AWWA M22 and most water utility design standards specify the "design" or "aged" C for capacity and pressure-zone analysis. Manufacturers' "C = 150" claim for plastic is for clean, new pipe under controlled conditions.

When to use Hazen-Williams vs. Darcy-Weisbach

Use Hazen-Williams when…	Use Darcy-Weisbach when…
Water at ~50–80°F	Other fluids, hot/cold extremes, viscous flow
Turbulent flow (typical mains)	Laminar / transitional flow (Re < 4000)
Fully-developed water-distribution pipes	Compressible flow, two-phase, complex networks needing energy balance
You need a quick spreadsheet calc	You're modeling in EPANET / WaterCAD / similar (those use D-W internally)

Sources: AWWA M11 (steel), M22 (sizing water service), M23 (PVC), M55 (HDPE). Mays, L.W. (2010). Water Distribution Systems Handbook. Hwang & Houghtalen, Fundamentals of Hydraulic Engineering Systems.

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Rational Method Runoff Coefficient (C) — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 15, 2026

For peak runoff: Q = C·i·A (US units — Q in cfs, i in in/hr, A in acres). The Rational Method is appropriate for drainage areas under ~200 acres, where the rainfall intensity is reasonably uniform and the hydrograph shape isn't needed. For larger watersheds or when you need volume / hydrograph routing, switch to the NRCS curve-number method.

By Surface Type

Surface	C (typical)	Range
Asphalt or concrete pavement	0.90	0.85–0.95
Brick / unit pavers, sealed joints	0.80	0.70–0.85
Roof, metal or membrane	0.95	0.90–0.95
Gravel, well-compacted	0.55	0.40–0.65
Gravel, loose	0.35	0.20–0.50
Bare clay soil, packed	0.55	0.40–0.65
Bare sandy soil, loose	0.20	0.10–0.30

By Land Use

Land use	C (typical)	Range
Downtown business district	0.85	0.70–0.95
Neighborhood business district	0.65	0.50–0.70
Light industrial	0.60	0.50–0.80
Heavy industrial	0.75	0.60–0.90
Residential, multifamily attached	0.65	0.60–0.75
Residential, multifamily detached	0.50	0.40–0.60
Residential, single-family (~¼ ac lots)	0.40	0.35–0.50
Residential, single-family (~½ ac lots)	0.30	0.25–0.40
Residential, large estate (1+ ac lots)	0.25	0.20–0.35
Parks, cemeteries	0.20	0.10–0.30
Playgrounds, school grounds	0.30	0.20–0.40
Railroad yards	0.30	0.20–0.40
Unimproved / open land	0.20	0.10–0.30

By Lawn / Soil / Slope

Lawn surface	Slope	C
Sandy soil, lawn	Flat (< 2%)	0.05–0.10
	Avg (2–7%)	0.10–0.15
	Steep (> 7%)	0.15–0.20
Heavy / clay soil, lawn	Flat (< 2%)	0.13–0.17
	Avg (2–7%)	0.18–0.22
	Steep (> 7%)	0.25–0.35

Frequency adjustment for high-return-period storms. The C values above are calibrated for ~10-year design storms. For larger storms, multiply C by C_f: C_f = 1.0 (T ≤ 10 yr), 1.10 (25 yr), 1.20 (50 yr), 1.25 (100 yr). Cap the product at 1.0. ASCE/WEF MOP-37, Table 5-2.

Composite (Weighted) C for a Mixed Site

For a site with multiple surfaces:

C_composite = Σ(C_i · A_i) / ΣA_i

Worked example for a 1.5-ac developed lot with 0.4 ac roof, 0.5 ac asphalt, 0.6 ac lawn (clay, flat):

C = (0.95·0.4 + 0.90·0.5 + 0.15·0.6) / 1.5
C = (0.380 + 0.450 + 0.090) / 1.5 = 0.61

Sources: ASCE/WEF MOP-37 (Design and Construction of Urban Stormwater Management Systems), 1992. NRCS TR-55. Wright-McLaughlin Engineers, Urban Storm Drainage Criteria Manual (UDFCD).

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NRCS Curve Number (CN) — TR-55 Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 15, 2026

Curve numbers for the NRCS / SCS runoff method: Q = (P − I_a)² / (P − I_a + S), with S = 1000/CN − 10 (in) and I_a = λS. Values below are ARC II (average antecedent condition) from TR-55 Table 2-2. For mixed-cover watersheds, area-weight the CN values directly — never weight runoff Q.

Urban / Developed Land Uses (TR-55 Table 2-2a)

Land use	% Imp.	A	B	C	D
Open space, poor cover (< 50% grass)	—	68	79	86	89
Open space, fair cover (50–75% grass)	—	49	69	79	84
Open space, good cover (> 75% grass)	—	39	61	74	80
Paved parking, roofs, driveways	100	98	98	98	98
Paved streets, curb & gutter	100	98	98	98	98
Paved streets, open ditches	—	83	89	92	93
Gravel roads	—	76	85	89	91
Dirt roads	—	72	82	87	89
Commercial / business	85	89	92	94	95
Industrial	72	81	88	91	93
Residential, 1/8-ac (townhouse)	65	77	85	90	92
Residential, 1/4-ac lots	38	61	75	83	87
Residential, 1/3-ac lots	30	57	72	81	86
Residential, 1/2-ac lots	25	54	70	80	85
Residential, 1-ac lots	20	51	68	79	84
Residential, 2-ac lots	12	46	65	77	82

Agricultural & Natural Land Uses (TR-55 Table 2-2c)

Land use / treatment	A	B	C	D
Fallow, bare soil	77	86	91	94
Row crops, straight, poor	72	81	88	91
Row crops, straight, good	67	78	85	89
Row crops, contoured, good	64	75	82	85
Small grain, straight, poor	65	76	84	88
Pasture, poor (< 50% cover)	68	79	86	89
Pasture, fair (50–75% cover)	49	69	79	84
Pasture, good (> 75% cover)	39	61	74	80
Meadow, continuous grass	30	58	71	78
Brush, fair	35	56	70	77
Woods, fair cover	36	60	73	79
Woods, good cover	30	55	70	77
Farmsteads (buildings, lanes)	59	74	82	86

Hydrologic Soil Groups (HSG)

Group	Soil texture	Min. infiltration	Runoff potential
A	Sand, loamy sand, sandy loam	> 0.30 in/hr	Low
B	Silt loam, loam	0.15–0.30 in/hr	Moderate
C	Sandy clay loam	0.05–0.15 in/hr	Moderately high
D	Clay, silty clay, shallow bedrock	< 0.05 in/hr	High

Site-specific HSG comes from NRCS Web Soil Survey (websoilsurvey.nrcs.usda.gov). Default to C or D when soils data is unavailable. For dual groups (A/D, B/D), the second letter applies to undrained conditions.

Antecedent runoff condition (ARC). Tabulated values are ARC II (average). Convert when required: CN_I = 4.2·CN_II / (10 − 0.058·CN_II) for dry; CN_III = 23·CN_II / (10 + 0.13·CN_II) for wet. Modern NRCS practice (NEH-630) uses ARC II for all design.

Composite (Weighted) CN for a Mixed Watershed

CN_composite = Σ(CN_i · A_i) / ΣA_i

Example — 50-ac watershed = 20 ac residential ½-ac (CN 70, HSG B) + 15 ac woods good (CN 55) + 15 ac pasture fair (CN 69):

CN = (20·70 + 15·55 + 15·69) / 50
CN = (1400 + 825 + 1035) / 50 = 65.2 → 65

Source: USDA NRCS (1986), Urban Hydrology for Small Watersheds (TR-55), Table 2-2. ARC conversion: NEH Part 630, Chapter 10. Values are ARC II, I_a = 0.2S basis.

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Weir Discharge Coefficients — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

All values for free-flow (non-submerged), well-ventilated nappe, with the head H measured upstream of the drawdown (typically 4·H upstream of the crest). Submergence corrections required when H_downstream/H_upstream > 0.6.

Sharp-Crested (Suppressed) Rectangular Weir

Q = C_d · L · H^3/2 (US units, Q in cfs, L & H in ft)

Source	C_d (US)	Notes
Rehbock formula (suppressed)	3.27 + 0.40·H/P	P = weir height; valid 0.03 ≤ H/P ≤ 1.0
Kindsvater-Carter (typical sharp-crested)	3.33	Common textbook design value
USBR (broad sharp-crest)	3.32–3.36

Side-contracted (Francis): subtract 0.1H per contraction from L → effective length L_e = L − 0.1nH (n = 1 or 2 contractions).

Broad-Crested Weir / Spillway

Q = C_d · L · H^3/2 (US units)

Crest geometry	C_d (US)	Notes
Square-edged broad-crest	2.6–3.1	varies with H/L_crest
Rounded upstream edge (r = 0.1L)	3.0–3.3
Ogee spillway (design head H_d)	3.95	maximum at design head; lower at partial heads
Ogee spillway (H = 0.5·H_d)	3.6
Roadway / parking lot overtopping	2.7–3.0	FHWA HEC-22, paved

V-Notch (Triangular) Weir — Free Discharge

Q = (8/15)·C_d·tan(θ/2)·√(2g)·H^5/2, often simplified to Q = K·H^5/2

Notch angle (θ)	C_d	K (US, Q in cfs, H in ft)
22.5°	0.611	0.497
30°	0.585	0.685
45°	0.581	1.035
60°	0.577	1.443
90° (most common)	0.578	2.49
120°	0.580	4.34

Valid for H > 0.2 ft and H/P < 0.4 (P = weir height above channel floor). Below H = 0.2 ft, surface tension errors dominate — use a reduced coefficient or a different measurement device.

Cipolletti (Trapezoidal) Weir

Side slopes 1H:4V, designed so contraction loss compensates for end contractions. Q = 3.367 · L · H^3/2, with C_d ≈ 0.63 (SI form).

Submergence: when downstream water matters. Once H_downstream/H_upstream > ~0.6, the weir is "submerged" and free-flow equations over-predict Q. Apply the Villemonte submergence correction: Q_submerged / Q_free = (1 − (H_d/H_u)ⁿ)^0.385, where n is the free-flow exponent (1.5 for rectangular, 2.5 for V-notch).

SI form (metric) of the same equations

Weir	Equation (SI: Q in m³/s, L & H in m)
Sharp-crested rect.	Q = (2/3) · C_d · L · √(2g) · H^3/2; C_d ≈ 0.62
Broad-crested	Q = C_d · L · √g · (2H/3)^3/2; C_d ≈ 0.85–1.0
V-notch (90°)	Q = (8/15) · 0.578 · tan(45°) · √(2g) · H^5/2 = 1.36 · H^5/2

Sources: USBR Water Measurement Manual 3rd ed. (2001), Chapter 7. Bos, M.G. (1989), Discharge Measurement Structures, ILRI Publication 20. Brater & King, Handbook of Hydraulics, 7th ed. USGS WSP 200 series.

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Time of Concentration Methods — Comparison

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

Time of concentration (T_c) is the time for runoff to travel from the hydraulically most distant point in the watershed to the design point. It controls the rainfall intensity used in the Rational Method (Q = CiA), and it sets the duration of the unit hydrograph in NRCS methods. Different methods make different assumptions about flow regime — the wrong choice can be off by 2× or more.

Decision matrix — pick the right method

Watershed type	Recommended method	Why
Rural, single-flow-path, A < 200 ac, slope 3–10%	Kirpich (1940)	Calibrated on rural watersheds in this exact range. Reasonable for steep grass / row-crop terrain.
Rural with significant overland flow before defined channel	Kerby (overland) + Kirpich (channel)	Kirpich underestimates pre-channelized travel time.
Mixed urban / rural, mixed flow regimes	NRCS TR-55 segmental	Splits into sheet flow (≤ 100 ft), shallow concentrated flow, channel flow. The standard for SWMP work.
Whole watershed lumped, NRCS hydrology	NRCS Lag formula	Built into TR-20 / HEC-HMS NRCS unit hydrograph. T_c = T_L / 0.6.
Airport, paved, very flat & short overland flow	FAA (1970)	Calibrated for runways. Short paved drainage paths.
Watershed too small or too large for any of the above	Don't use Tc — use a different model	For A < 5 ac use direct sheet-flow time; for A > 2,000 ac use a routing model.

The equations, side-by-side (US units)

Kirpich (1940)

T_c (min) = 0.0078 · L^0.77 · S^−0.385

L = flow length (ft); S = average watershed slope (ft/ft). Multiply by 0.4 for paved overland flow, 0.2 for asphalt/concrete channels. Validity: 1 to 200 ac, slope 3–10%.

NRCS Lag (SCS)

T_L (hr) = L^0.8(S+1)^0.7 / (1900 · Y^0.5)

T_c = T_L / 0.6

L = hydraulic length (ft); S = (1000/CN) − 10 (potential maximum retention, in); Y = average watershed slope (%). Validity: A < 2,000 ac, CN-based runoff hydrology.

TR-55 Segmental (3-segment travel time)

Sheet flow (≤ 100 ft, smooth surfaces only):

T_t,sheet = 0.007 · (n · L)^0.8 / (P₂^0.5 · S^0.4)

Shallow concentrated flow: V = 16.1345·√S (unpaved), V = 20.3282·√S (paved); T_t = L / (3600·V)

Channel flow: Manning's equation V = (1.486/n) · R^2/3 · S^1/2; T_t = L / (3600·V)

T_c = T_t,sheet + T_t,shallow + T_t,channel

Kerby / Hathaway (overland)

T_c (min) = 0.83 · (L · n / √S)^0.467

For overland flow only, L ≤ 1200 ft. n is a retardance coefficient (0.02 paved, 0.10 grass, 0.40 woods).

FAA

T_c (min) = 1.8 · (1.1 − C) · L^0.5 / S^0.333

C = rational-method runoff coefficient; L in ft; S in %.

Floor your T_c at 5 minutes for the Rational Method. NOAA Atlas 14 and most IDF curves don't extrapolate reliably below 5 min. Many regulators (NCDOT, FHWA HEC-22, UDFCD) require a 5-min minimum. A computed T_c of 2 min on a tiny urban catchment will give you an absurd intensity from the IDF curve — use 5 min instead.

Common mistakes

Using Kirpich on urban watersheds. Kirpich was calibrated on rural Tennessee farmland. Apply the 0.4 / 0.2 multipliers if any portion is paved, or just switch to TR-55 segmental.
Ignoring the 100-ft sheet-flow cap in TR-55. Sheet flow transitions to shallow concentrated flow within ~100 ft on natural terrain (50 ft on flat or vegetated terrain). Carrying sheet flow longer over-estimates Tc.
Mixing methods on one path. If you start with Kirpich for a rural reach and then use TR-55 sheet-flow on the same upstream segment, you've double-counted. Pick one method per flow segment.
Forgetting the lag-to-Tc conversion. NRCS lag (T_L) is not Tc — it's 60% of it. T_c = T_L / 0.6.

Sources: Kirpich, P.Z. (1940), Civil Engineering, Vol. 10, p. 362. NRCS TR-55 (1986). USDA NEH Part 630, Chapter 15. FAA AC 150/5320-5C. McCuen (2017), Hydrologic Analysis & Design, 4th ed.

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Minor Loss Coefficients (K) — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 15, 2026

Localized losses from fittings, valves, bends, entrances and exits, by the velocity-head method: h_L = K·V²/(2g), with V the mean pipe velocity. Values are typical for fully-turbulent flow; K varies with size, Reynolds number, and manufacturer — use Crane TP-410 or vendor data for final design.

Entrances & Exits

Component	K
Re-entrant / projecting pipe entrance	0.8–1.0
Sharp-edged (flush) entrance	0.5
Slightly rounded entrance (r/D ≈ 0.1)	0.2–0.25
Well-rounded entrance (r/D ≥ 0.15)	0.04–0.05
Exit (pipe → reservoir), any shape	1.0

Bends & Elbows

Component	Threaded	Flanged
90° elbow, regular (standard)	0.9	0.3
90° elbow, long radius	0.6	0.2
45° elbow, regular	0.4	0.2
180° return bend	1.5	0.2
90° bend, smooth (r/D = 4–6)	0.15–0.30
Mitered bend, 90° (no vanes)	1.1–1.3

Tees

Component	Threaded	Flanged
Tee, line (run-through) flow	0.9	0.2
Tee, branch flow	2.0	1.0

Valves (Fully Open)

Valve type	K
Ball valve, full bore	0.05
Gate valve	0.15–0.2
Butterfly valve	0.3–1.2
Swing check valve	2.0–2.5
Angle valve	2–5
Globe valve	6–10
Foot valve with strainer (hinged)	0.8–1.5

Gate valves throttle steeply: ¾ open ≈ K 1.0–1.2, ½ open ≈ 5.6, ¼ open ≈ 17+. Always design for the fully-open value unless throttling is intended.

Sudden Expansion & Contraction

Component	K (based on smaller-pipe velocity)
Sudden expansion	K = (1 − A₁/A₂)² = (1 − (d₁/d₂)²)²
Sudden contraction	K ≈ 0.5(1 − A₂/A₁) (0 for no change, 0.5 for exit into pipe)

Combine with friction loss. Total head loss over a run is h = (ΣK + f·L/D)·V²/(2g). For long pipelines, minor losses are often negligible; for short runs with many fittings they can dominate. Equivalent-length alternative: L_eq = K·D/f added to the pipe length.

Sources: Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe (Technical Paper 410). Munson et al., Fundamentals of Fluid Mechanics, Table 8.2. Values are representative; manufacturer/size variation is significant for valves.

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Pipe Absolute Roughness (ε) — Darcy-Weisbach Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 15, 2026

Absolute roughness ε sets the relative roughness ε/D for the Moody diagram and the friction factor f in h_f = f·(L/D)·V²/(2g). Values below are for clean, new pipe unless noted. For aging water mains, design to an end-of-life (aged) roughness.

Absolute Roughness by Material

Material	ε (mm)	ε (ft)
Drawn tubing, glass, brass, copper	0.0015	0.000005
PVC, HDPE, smooth plastic	0.0015–0.007	0.000005–0.000023
Commercial steel / wrought iron (new)	0.045	0.00015
Asphalted cast iron	0.12	0.0004
Galvanized iron	0.15	0.0005
Ductile iron, cement-mortar lined	0.10–0.12	0.00033–0.0004
Cast iron (uncoated, new)	0.26	0.00085
Wood stave	0.18–0.9	0.0006–0.003
Concrete (smooth to rough)	0.3–3.0	0.001–0.01
Riveted steel	0.9–9.0	0.003–0.03
Corrugated metal pipe (annular)	~45	~0.15
Aged / tuberculated steel or cast iron	1.0–3.0	0.003–0.01

Friction Factor Equations (Turbulent, Re > 4000)

Colebrook-White (implicit, the Moody-diagram basis):

1/√f = −2 log₁₀( ε/(3.7D) + 2.51/(Re·√f) )

Swamee-Jain (explicit, ±1% over 4000 < Re < 10⁸, 10⁻⁶ < ε/D < 10⁻²):

f = 0.25 / [ log₁₀( ε/(3.7D) + 5.74/Re^0.9 ) ]²

Laminar flow (Re < 2000), roughness irrelevant:

f = 64/Re

Flow regimes. Re < 2000 laminar (f = 64/Re); 2000–4000 transitional (avoid for design); > 4000 turbulent (use Colebrook / Swamee-Jain). Re = ρVD/μ = VD/ν. For water at 60°F, ν ≈ 1.21×10⁻⁵ ft²/s (1.12×10⁻⁶ m²/s).

Sources: Moody, L.F. (1944), "Friction Factors for Pipe Flow," Trans. ASME. White, F.M., Fluid Mechanics, Table 6.1. Swamee, P.K. & Jain, A.K. (1976), J. Hydraulics Div., ASCE. Aged values: AWWA M11 / utility practice.

Card: pe-calc.com/cheat-sheets/pipe-roughness-darcy

Culvert Entrance Loss Coefficients (K_e) — HDS-5

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

Entrance loss under outlet control: h_e = K_e·V²/(2g), with V the barrel velocity. K_e depends on inlet geometry and barrel material. Values are from FHWA HDS-5, Table 12 (Hydraulic Design of Highway Culverts).

Concrete Pipe

Inlet configuration	K_e
Projecting from fill, socket (groove) end	0.2
Projecting from fill, square-cut end	0.5
Headwall / headwall & wingwalls, socket end	0.2
Headwall / headwall & wingwalls, square edge	0.5
Headwall, rounded edge (radius ≈ 1/12 D)	0.2
Beveled edges (33.7° or 45° bevels)	0.2
Side- or slope-tapered inlet	0.2

Corrugated Metal Pipe (CMP)

Inlet configuration	K_e
Projecting from fill (no headwall)	0.9
Mitered to conform to fill slope	0.7
Headwall or headwall & wingwalls, square edge	0.5
End section conforming to fill slope	0.5
Beveled edges (33.7° or 45° bevels)	0.25
Side- or slope-tapered inlet	0.2

Box Culvert (Reinforced Concrete)

Inlet configuration	K_e
Wingwalls 30°–75° to barrel, square edge at crown	0.4
Wingwalls 30°–75°, crown edge rounded (r ≈ 1/12 D)	0.2
Wingwalls 90° & 15° to barrel, square edge	0.5
Wingwalls parallel (extension of sides), square edge	0.7
Beveled edges on 3 sides	0.2

Outlet-control headwater.

HW = TW + H − L·S_o, H = (1 + K_e + 29n²L/R^1.33)·V²/(2g)

where H is total head loss (entrance + friction + exit), TW is tailwater depth, L barrel length, S_o barrel slope, R hydraulic radius. The "1" is the exit (velocity-head) loss. Design HW is the larger of the inlet-control and outlet-control results.

Source: FHWA, Hydraulic Design of Highway Culverts (HDS-5, Publication FHWA-HIF-12-026), Table 12. Inlet-control headwater uses the nomographs or the K, M, c, Y coefficients of HDS-5 Table 9 / Appendix A, not K_e.

Card: pe-calc.com/cheat-sheets/culvert-entrance-loss

NRCS / SCS 24-Hour Storm Distributions

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

The synthetic 24-hour rainfall distributions used to turn a design-storm depth into a hydrograph (TR-20, TR-55, HEC-HMS, HydroCAD). Each is a dimensionless mass curve — fraction of total 24-hour rainfall vs. time — chosen by region.

The Four Distributions

Type	Climate / character	Peak intensity
IA	Pacific maritime, mild frontal storms (least intense)	Lowest
I	Pacific maritime, slightly more intense than IA	Low
III	Gulf / Atlantic coastal, tropical & hurricane systems	High
II	Continental US, intense short-duration convective burst	Highest

All four place the peak rainfall intensity near the 12-hour (midpoint) mark; they differ in how sharply the rain is concentrated around that peak. Type II has the steepest central burst, Type IA the flattest.

Geographic Applicability

Type	Where it applies
IA	Coastal Pacific Northwest (coastal OR, WA), northern coastal CA
I	California, western OR/WA (inland of IA zone), Hawaii, Alaska
II	Most of the continental US — the default for the interior and East outside the coastal strip
III	Gulf Coast and Atlantic coastal strip: Florida, coastal TX, LA, MS, AL, GA, and the Carolinas to the Delmarva

NOAA Atlas 14 is superseding the Type curves. NRCS now provides regional dimensionless distributions derived from NOAA Atlas 14 that better fit local rainfall statistics, and a number of state stormwater manuals now require them. Confirm the current state / NRCS requirement before defaulting to Type II/III — the legacy types remain valid only where a regional distribution has not been adopted.

How It Is Used

Pick the 24-hour depth from NOAA Atlas 14 for the design return period (e.g., 25-yr or 100-yr), select the regional distribution, and run it through the NRCS unit hydrograph in TR-20 / HEC-HMS / HydroCAD. The distribution shape controls peak discharge: the same 24-hour depth on a Type II curve yields a higher peak than on a Type IA curve for the same watershed.

Sources: USDA NRCS, TR-55, Urban Hydrology for Small Watersheds (1986), Appendix B; NRCS NEH Part 630, Chapter 4; NOAA Atlas 14 (hdsc.nws.noaa.gov). Regional distribution adoption varies by state.

Card: pe-calc.com/cheat-sheets/scs-storm-distributions

Open Channel Geometry — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

Section properties for Manning's and critical-flow analysis. A = flow area, P = wetted perimeter, R = A/P = hydraulic radius, T = top width, D = A/T = hydraulic depth. Side slope z is horizontal:vertical (z H : 1 V), depth y, bottom width b.

Section Property Formulas

Shape	Area A	Wetted perim. P	Hyd. radius R	Top width T
Rectangular	b·y	b + 2y	by/(b+2y)	b
Trapezoidal	(b + zy)y	b + 2y√(1+z²)	A/P	b + 2zy
Triangular	z·y²	2y√(1+z²)	zy / 2√(1+z²)	2zy
Circular*	(D&sub0;²/8)(θ − sinθ)	D&sub0;θ/2	(D&sub0;/4)(1 − sinθ/θ)	D&sub0; sin(θ/2)

*Circular (partly full), diameter D&sub0;, flow depth y: θ = 2·arccos(1 − 2y/D&sub0;) radians. Full pipe: A = πD&sub0;²/4, R = D&sub0;/4.

Manning's Equation

Q = (k/n) A R^2/3 S^1/2 k = 1.486 (US, ft) / 1.0 (SI, m)

V = Q/A = (k/n) R^2/3 S^1/2. S is the channel (friction) slope; n is Manning's roughness. Normal depth is found by solving Manning's for the y that passes the design Q — iterative for all shapes except the simplest.

Best Hydraulic Section (Max Q for Given Area)

Shape	Optimum condition	R at optimum
Rectangular	b = 2y (width = twice depth)	y/2
Trapezoidal	Half-hexagon: z = 1/√3 (60° sides)	y/2
Triangular	z = 1 (90° vee, sides at 45°)	y/(2√2)
Semicircle	Overall most efficient open shape	y/2

R vs. D — don't mix them up. Use R = A/P (hydraulic radius) in Manning's / Chézy resistance. Use D = A/T (hydraulic depth) in the Froude number Fr = V/√(gD) and critical-depth checks. They coincide only for a wide rectangular channel (R ≈ D ≈ y as b → ∞).

Sources: Chow, V.T. (1959), Open-Channel Hydraulics, Table 2-1. Sturm, T.W., Open Channel Hydraulics. Standard prismatic-channel geometry.

Card: pe-calc.com/cheat-sheets/open-channel-geometry

Orifice Discharge Coefficients (C_d) — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

For outlet structures, detention risers, and tank drains: Q = C_d·A·√(2gh), with A the orifice area and h the head on the centroid (free) or the head difference (submerged). C_d = C_c·C_v (contraction × velocity).

Discharge Coefficient by Orifice Type

Orifice / outlet type	C_d
Sharp-edged (thin plate) orifice	0.61–0.62
Standard stormwater orifice outlet (design value)	0.60
Rounded / bell-mouth entrance	0.95–0.98
Short tube (L ≈ 2–3 diameters), flowing full	0.80–0.82
Borda re-entrant tube, running full	~0.72
Submerged orifice (use head difference)	~0.61
Pipe/culvert entrance treated as orifice	0.60–0.62

Component Coefficients (Sharp-Edged)

Coefficient	Value	Meaning
Contraction C_c	~0.62	vena-contracta area / orifice area
Velocity C_v	~0.98	actual / ideal jet velocity
Discharge C_d	~0.61	C_c × C_v

Weir-to-orifice transition. A riser opening behaves as a weir at low head (Q ∝ h^1.5) and as an orifice once submerged (Q ∝ h^0.5). Size the outlet by computing both at each stage and taking the lower discharge. The flatter orifice curve is why small orifices control low-flow release rates.

Free orifice: Q = C_d A √(2gh) · Submerged: Q = C_d A √(2gΔh)

Sources: Brater & King, Handbook of Hydraulics. ASCE/WEF MOP-77. Standard stormwater outlet-structure practice (Cd = 0.6 for orifice plates).

Card: pe-calc.com/cheat-sheets/orifice-discharge-coefficients

Hydraulic Jump — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

The rapid transition from supercritical to subcritical flow used to dissipate energy below spillways, chutes, and outlets. Subscript 1 = upstream (supercritical) section, 2 = downstream (subcritical) section.

Core Equations (Rectangular Channel)

Froude: Fr&sub1; = V&sub1; / √(g·y&sub1;)

Sequent depth: y&sub2;/y&sub1; = ½( √(1 + 8·Fr&sub1;²) − 1 )

Energy loss: ΔE = (y&sub2; − y&sub1;)³ / (4·y&sub1;·y&sub2;)

Jump length: L_j ≈ 6·y&sub2; (range 4.5–7·y&sub2;)

The relation is reversible: y&sub1;/y&sub2; = ½(√(1 + 8·Fr&sub2;²) − 1). The two depths are conjugate (equal specific force), not equal specific energy — the difference is the dissipated ΔE.

Jump Classification by Upstream Froude Number

Fr&sub1;	Type	Character & energy dissipation
1.0–1.7	Undular	Standing waves, minimal loss (< 5%)
1.7–2.5	Weak	Smooth surface, low loss (5–15%)
2.5–4.5	Oscillating	Avoid — jet oscillates, sends damaging waves downstream (15–45%)
4.5–9.0	Steady	Stable, well-balanced, best performance (45–70%)
> 9.0	Strong	Rough, intense turbulence, very effective (up to ~85%)

Design targets. Aim the design point into the steady jump range (Fr&sub1; 4.5–9). The oscillating range (2.5–4.5) is functional but its surface waves erode unlined channels and riprap downstream — USBR Type I–IV stilling basins add chute blocks, baffle piers, and end sills to stabilize the jump and shorten the basin in this range.

Sources: Chow, V.T. (1959), Open-Channel Hydraulics. USBR, Hydraulic Design of Stilling Basins and Energy Dissipators (Engineering Monograph 25). FHWA HEC-14.

Card: pe-calc.com/cheat-sheets/hydraulic-jump

Riprap Sizing Methods — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 15, 2026

Sizing stone for channel, bank, and outlet protection. The design quantity is the median stone size D50, set by velocity or shear; gradation, layer thickness, and an underlying filter complete the design.

Velocity-Based D50 (Isbash)

D50 = V² / [ C² · 2g · (S_s − 1) ]

Term	Meaning / typical value
V	Design (local) velocity at the stone
S_s	Stone specific gravity, ~2.65
C	Isbash coefficient: ~0.86 high-turbulence, ~1.20 low-turbulence
g	32.2 ft/s² (9.81 m/s²)

D50 Riprap Size Chart by Velocity (Isbash, S_s = 2.65)

Velocity V (ft/s)	D50, high turbulence (C = 0.86)	D50, low turbulence (C = 1.20)
4	0.20 ft (2.4 in)	0.10 ft (1.3 in)
6	0.46 ft (5.5 in)	0.24 ft (2.8 in)
8	0.81 ft (9.8 in)	0.42 ft (5.0 in)
10	1.27 ft (15.3 in)	0.65 ft (7.8 in)
12	1.83 ft (22.0 in)	0.94 ft (11.3 in)
14	2.49 ft (29.9 in)	1.28 ft (15.4 in)
16	3.26 ft (39.1 in)	1.67 ft (20.1 in)

SI: 1 m/s → 42 / 21 mm; 2 m/s → 167 / 86 mm; 3 m/s → 376 / 193 mm; 4 m/s → 668 / 343 mm (high / low turbulence). Computed from the equation above with g = 32.2 ft/s² (9.81 m/s²). D50 scales with V², so doubling velocity quadruples stone size.

This is a screening chart, not a design. Isbash uses velocity alone. It has no flow depth, side-slope or bend correction, so on banks and bends the USACE EM 1110-2-1601 (Maynord) or HEC-11 size usually governs. The riprap calculator runs both and reports the larger D50.

FHWA HEC-11 (revetment) and HEC-23 (bridge/abutment countermeasures) give velocity- and depth-based D50 relations with bank-angle and specific-gravity corrections; use the method your reviewer requires.

Gradation Limits (Well-Graded Riprap)

Ratio	Target
D85 / D15 (uniformity)	1.5–2.5 (up to ~4.6 allowed)
D100 (max) / D50	≤ 2.0
D50 / D15	~1.5–2.0

Well-graded stone interlocks and resists displacement; uniform (gap-graded) stone is unstable and prone to winnowing.

Layer Thickness & Filter

Item	Rule
Blanket thickness	≥ larger of 1.5·D50 or D100; increase for steep / submerged placement
Granular filter	Terzaghi: D15_f/D85_b < 4–5 < D15_f/D15_b
Geotextile filter	Alternative to granular; AOS sized to retain base soil

Always place a filter. Riprap without a filter fails by piping — turbulence pulls base fines through the voids and the blanket settles and unravels. Granular filter or geotextile is not optional on erodible subgrade.

Sources: FHWA HEC-11 Design of Riprap Revetment; FHWA HEC-23 Bridge Scour and Stream Instability Countermeasures; USACE EM 1110-2-1601; Isbash (1936). State DOT riprap classes (e.g., NCDOT Class A/B/1/2) define standard gradations — check the governing spec.

Card: pe-calc.com/cheat-sheets/riprap-sizing-methods

Specific Energy & Critical Depth — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

Specific energy is energy per unit weight relative to the channel bottom. It controls open-channel transitions, controls, and the sub/supercritical state of the flow.

Core Equations

Specific energy: E = y + V²/(2g) = y + Q²/(2gA²)

Critical condition (general): Q²T / (gA³) = 1 ⇔ Fr = 1

Rectangular: y_c = (q²/g)^1/3, q = Q/b, E_min = 1.5·y_c, V_c = √(g·y_c)

Flow State by Depth

Condition	State	Character
y > y_c (Fr < 1)	Subcritical	Tranquil, deep/slow; controls act from downstream
y = y_c (Fr = 1)	Critical	Minimum E; unstable, used as a flow-measurement control
y < y_c (Fr > 1)	Supercritical	Rapid, shallow/fast; controls act from upstream

Alternate Depths

For any E > E_min, two alternate depths pass the same Q at the same specific energy — one subcritical, one supercritical. They are equal in specific energy. Do not confuse them with the conjugate (sequent) depths of a hydraulic jump, which are equal in specific force (momentum) and dissipate energy between them.

Why it matters. Critical depth is a control section — at a free overfall, a broad-crested weir, or the crest of a bump, flow passes through y_c and the stage-discharge relation is fixed. Knowing whether your channel runs sub- or supercritical determines where backwater/drawdown profiles start and whether a hydraulic jump can form.

Source: Chow, V.T. (1959), Open-Channel Hydraulics, Ch. 3. Henderson, F.M., Open Channel Flow.

Card: pe-calc.com/cheat-sheets/specific-energy-critical-depth

Storm Sewer Design Criteria — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

Common design criteria for gravity storm sewers. Values are typical of state and municipal standards (e.g., GLUMRB "Ten States Standards" lineage) — always confirm against the governing local spec, which controls.

Velocity & Slope

Criterion	Typical value
Minimum full-flow velocity (anti-deposition)	2.0–2.5 ft/s
Maximum velocity (abrasion / scour)	10–15 ft/s
Minimum slope	set to achieve 2 ft/s full
Design flow condition	full or just-full (no surcharge)

Manning's n (Storm Sewer Pipe)

Pipe material	n (design)
Concrete / reinforced concrete pipe (RCP)	0.013
PVC / HDPE (smooth interior)	0.012–0.013
Corrugated metal pipe (CMP), annular	0.024
CMP, helical (small dia.)	0.012–0.024
Ductile iron, cement-lined	0.013

Many agencies require n = 0.013 even for plastic pipe, to account for joints, debris, and aging. Use the value the reviewer mandates.

Size, Cover & Storm

Criterion	Typical value
Minimum trunk pipe diameter	12–15 in
Minimum cover over pipe	1–3 ft (loading / frost dependent)
Design storm (local sewers)	10-yr (25/100-yr check)
Design storm (trunk / outlets)	25–100-yr

Full-flow capacity (Manning's).

Q_full = (1.49/n)·(πD²/4)·(D/4)^2/3·S^1/2

Maximum capacity actually occurs at ~93% depth (not full); peak velocity at ~81% depth. For partly-full analysis use the circular-section geometry with the central angle θ.

Sources: GLUMRB, Recommended Standards for Wastewater Facilities ("Ten States Standards"); ASCE/WEF MOP-37; typical state DOT and municipal storm-drainage manuals. Criteria vary by jurisdiction.

Card: pe-calc.com/cheat-sheets/storm-sewer-design-criteria

Stokes Settling Velocity — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

Terminal settling velocity of discrete particles, and the surface-overflow-rate basis for sedimentation basins, forebays, and grit chambers.

Settling Velocity by Regime

Stokes (Re < 1): v_s = g(ρ_s − ρ)d² / (18μ)

Newton (Re > 1000): v_s ≈ 1.74·√( g(S_s − 1)d )

Regime	Particle Re	Applies to
Stokes (laminar)	< 1	Silt, clay, fine sand
Transition	1–1000	Medium–coarse sand
Newton (turbulent)	> 1000	Gravel, large grit

Particle Re = ρ·v_s·d/μ. S_s = ρ_s/ρ (quartz sand ≈ 2.65). Water at 20°C: μ = 1.00×10⁻³ Pa·s, ρ = 998 kg/m³.

Typical Settling Velocities (quartz, 20°C water)

Particle	Diameter	v_s (approx.)
Coarse sand	1.0 mm	~100 mm/s
Medium sand	0.5 mm	~50 mm/s
Fine sand	0.1 mm	~6–8 mm/s
Very fine sand	0.05 mm	~2 mm/s
Silt	0.01 mm	~0.08 mm/s
Clay	0.001 mm	~0.0008 mm/s

Fine silt/clay settle far too slowly for practical basins — they require flocculation or filtration. The d² dependence is why removing the fine fraction is so hard.

Surface overflow rate governs removal.

v_o = Q / A_s

In ideal Type I settling, every particle with v_s ≥ v_o is fully captured — removal depends on surface area, not depth. Size sedimentation basins and forebays by overflow rate, then check detention time and short-circuiting.

Sources: Camp, T.R. (1946), "Sedimentation and the Design of Settling Tanks," Trans. ASCE. Metcalf & Eddy, Wastewater Engineering. Reynolds & Richards, Unit Operations and Processes in Environmental Engineering.

Card: pe-calc.com/cheat-sheets/stokes-settling-velocity

Reynolds Number & Flow Regimes — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

The ratio of inertial to viscous forces — the single number that decides whether flow is laminar or turbulent, and therefore which friction relation applies.

Definition

Re = ρVL/μ = VL/ν (ν = μ/ρ)

Characteristic length L: pipe → diameter D (full-flow); open channel → hydraulic radius R = A/P. (Some texts use 4R for channels to align with the pipe definition — check which convention a threshold assumes.)

Flow Regime Thresholds

System	Laminar	Transitional	Turbulent
Pipe flow (L = D)	< 2100	2100–4000	> 4000
Open channel (L = R)	< 500	500–2000	> 2000
Flow around a sphere/particle	< 1	1–1000	> 1000

Kinematic Viscosity of Water

Temp	ν (m²/s)	ν (ft²/s)
5°C / 41°F	1.52×10⁻⁶	1.63×10⁻⁵
10°C / 50°F	1.31×10⁻⁶	1.41×10⁻⁵
20°C / 68°F	1.00×10⁻⁶	1.08×10⁻⁵
30°C / 86°F	0.80×10⁻⁶	0.86×10⁻⁵

Need density, specific weight or vapor pressure too? See the full water properties table (0–100°C / 32–212°F).

Why it matters. In laminar pipe flow the friction factor is simply f = 64/Re (roughness irrelevant). In turbulent flow you need Colebrook / Swamee-Jain with the relative roughness. The Froude number (not Reynolds) governs the sub/supercritical state of open-channel flow — Reynolds only tells you it's turbulent, which it nearly always is.

Sources: White, F.M., Fluid Mechanics. Munson et al., Fundamentals of Fluid Mechanics. Viscosity values: standard water property tables.

Card: pe-calc.com/cheat-sheets/reynolds-number-regimes

Pipe Fitting Equivalent Lengths — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

Equivalent length of straight pipe for fittings and valves: L_eq = (L/D)·D. Companion to the K-coefficient card — the two methods are related by K = f·(L/D).

L/D Ratios (Crane TP-410, fully turbulent)

Component	L/D
90° elbow, standard	30
90° elbow, long radius	16
45° elbow, standard	16
180° return bend	50
Tee, flow through run	20
Tee, flow through branch	60
Gate valve, fully open	8
Ball valve, full port	3
Butterfly valve (8″ and smaller)	≈ 45
Swing check valve	100
Lift check valve	600
Globe valve, fully open	340
Angle valve, fully open	150

Equivalent Length in Feet of Straight Pipe

Component	2″	4″	6″	8″	12″
90° elbow, standard (30)	5.0	10	15	20	30
90° elbow, long radius (16)	2.7	5.3	8.0	11	16
45° elbow (16)	2.7	5.3	8.0	11	16
Tee, run (20)	3.3	6.7	10	13	20
Tee, branch (60)	10	20	30	40	60
Gate valve (8)	1.3	2.7	4.0	5.3	8.0
Swing check (100)	17	33	50	67	100
Globe valve (340)	57	113	170	227	340

L_eq (ft) = (L/D) × d(in)/12, using nominal size as an approximation of inside diameter. For precise work use the actual ID of the pipe schedule.

Converting between methods.

K = f · (L/D) ↔ L_eq = K · D / f

With f ≈ 0.019 (typical clean water, mid-size pipe), a standard 90° elbow: K = 0.019 × 30 ≈ 0.57 by strict conversion — but tabulated K values (≈ 0.3 flanged) are measured, not converted. Small discrepancies between the two tables are real and expected; stay within one method for a given calculation.

Worked Example — 6″ suction line

Component	Qty	L_eq each (ft)	Total (ft)
90° elbow, long radius	3	8.0	24
Gate valve	1	4.0	4
Tee, run	1	10	10
ΣL_eq			38

Actual run of 60 ft → hydraulic length 60 + 38 = 98 ft. The fittings add over 60% to the friction loss of this short run — typical of station piping.

Sources: Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe (Technical Paper 410), equivalent-length tables. NFPA 13 (fire protection uses its own similar tables). Values representative for screwed/welded steel; manufacturer data governs for critical valves.

Card: pe-calc.com/cheat-sheets/equivalent-lengths

Unit Weights of Construction Materials — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

Dead-load and geotechnical unit weights in US customary (pcf) and SI (kN/m³). Conversion: 1 pcf = 0.1571 kN/m³.

Water & Fluids

Material	pcf	kN/m³
Fresh water	62.4	9.81
Seawater	64.0	10.05
Ice	57	9.0

Structural Materials

Material	pcf	kN/m³
Concrete, plain (normalweight)	145	22.8
Concrete, reinforced	150	23.6
Concrete, structural lightweight	90–115	14–18
Steel	490	77.0
Aluminum	165–170	26–27
Timber, softwood (SYP, DF)	32–37	5.0–5.8
Timber, hardwood	40–47	6.3–7.4
Brick masonry	120–130	19–20
CMU, grouted solid (normalweight)	135–140	21–22
Asphalt pavement (HMA)	140–145	22–23
Glass	160	25.1

Soils & Aggregates (moist unless noted)

Material	pcf	kN/m³
Sand, loose, dry	90–110	14–17
Sand, compacted	110–130	17–20
Sand, saturated	115–135	18–21
Clay, soft	100–120	16–19
Clay, stiff	115–135	18–21
Silt	100–125	16–20
Gravel / sandy gravel, compacted	120–140	19–22
Crushed stone base (compacted)	125–145	20–23
Structural fill (typ. design value)	120	18.9
Rock, solid (granite, limestone)	160–170	25–27
Riprap / gabion stone, bulk (30–40% voids)	100–120	16–19
Topsoil	80–100	13–16

Buoyant unit weight.

γ′ = γ_sat − γ_w

Below the water table, effective (buoyant) unit weight drives effective stress: saturated sand at 125 pcf has γ′ = 125 − 62.4 ≈ 63 pcf — half the total. Using total instead of buoyant weight below the water table is one of the classic geotechnical calculation errors.

Sources: ASCE/SEI 7-22, Table C3.1-1a (materials). Das, B.M., Principles of Geotechnical Engineering (soils). Terzaghi & Peck, Soil Mechanics in Engineering Practice. Ranges are handbook screening values; project-specific data governs final design.

Card: pe-calc.com/cheat-sheets/unit-weights

Presumptive Bearing Values — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

Allowable foundation pressures by soil class per IBC Table 1806.2, for use where the code permits design without a site-specific geotechnical investigation. Confirm local amendments — several states modify this table.

IBC Table 1806.2 — Presumptive Load-Bearing Values

Class of material	Vertical bearing (psf)	Lateral bearing (psf/ft depth)	Lateral sliding
1. Crystalline bedrock	12,000	1,200	μ = 0.70
2. Sedimentary & foliated rock	4,000	400	μ = 0.35
3. Sandy gravel and/or gravel (GW, GP)	3,000	200	μ = 0.35
4. Sand, silty sand, clayey sand, silty gravel, clayey gravel (SW, SP, SM, SC, GM, GC)	2,000	150	μ = 0.25
5. Clay, sandy clay, silty clay, clayey silt, silt, sandy silt (CL, ML, MH, CH)	1,500	100	cohesion 130 psf

Sliding: friction coefficient μ applies to dead load for classes 1–4; for class 5, cohesion times contact area, not to exceed one-half the dead load. Mud, organic silt, organic clays, peat, and unprepared fill have no presumptive capacity — investigation required (IBC 1806.2 footnotes).

Presumptive means "absent better information." These values are screening-level and deliberately low. A geotechnical investigation routinely doubles or triples usable bearing on decent soils, pays for itself on anything bigger than light-frame construction, and is mandatory where soils are questionable, expansive, or saturated (IBC 1803). The values here are the floor, not the target.

Quick Screening Numbers

Situation	Working number
Continuous wall footing, 1,500 psf soil, 8-ft wall	≈ 16″–24″ wide
Column on 2,000 psf soil, 40 kip service load	≈ 4.5 ft × 4.5 ft
Minimum footing depth (frost, typical mid-Atlantic/Southeast)	12″–36″ per local code

Screening only — size from actual loads with the bearing capacity tool or the geotech report's allowable pressure.

Sources: 2021 International Building Code, §1806 and Table 1806.2 (ICC). Soil classifications per ASTM D2487 (USCS). Check state/local amendments — the table is frequently modified.

Card: pe-calc.com/cheat-sheets/presumptive-bearing

Rebar Development Length — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

Straight tension development, ACI 318-19 simplified method (§25.4.2.3). Grade 60, normalweight concrete, uncoated bars, favorable spacing/cover case. Minimum l_d = 12 in.

l_d = f_y·Ψ_t·Ψ_e / (25·λ·√f′_c) · d_b (≤ #6) | f_y·Ψ_t·Ψ_e / (20·λ·√f′_c) · d_b (≥ #7)

Bottom Bars, Grade 60 — l_d in inches (rounded up)

Bar	d_b (in)	f′_c = 3,000	f′_c = 4,000	f′_c = 5,000
#3	0.375	17	15	13
#4	0.500	22	19	17
#5	0.625	28	24	22
#6	0.750	33	29	26
#7	0.875	48	42	38
#8	1.000	55	48	43
#9	1.128	62	54	48
#10	1.270	70	61	54
#11	1.410	78	67	60

In bar diameters: 43.8/37.9/33.9·d_b (≤ #6) and 54.8/47.4/42.4·d_b (≥ #7) at 3,000/4,000/5,000 psi respectively. Note the jump from #6 to #7 — the code's two-tier equation, not a typo.

Modifiers

Condition	Factor
Top bar (> 12″ fresh concrete below), Ψ_t	× 1.3
Epoxy-coated, cover < 3d_b or spacing < 6d_b, Ψ_e	× 1.5
Epoxy-coated, other, Ψ_e	× 1.2
Lightweight concrete, λ = 0.75	× 1.33
Simplified-method unfavorable spacing/cover case	× 1.5
Class B tension lap splice	1.3 × l_d

Ψ_t·Ψ_e need not exceed 1.7. Favorable case requires: clear spacing ≥ d_b, clear cover ≥ d_b, and code-minimum stirrups over l_d — or clear spacing ≥ 2d_b with cover ≥ d_b.

Sharper answers exist. The detailed equation of ACI 318-19 §25.4.2.4 with the confinement index (c_b + K_tr)/d_b can cut these lengths meaningfully when cover and transverse steel are generous, and hooked or headed bars (§25.4.3–25.4.4) develop in far less length where straight embedment doesn't fit. This card is the simplified upper-bound method used for quick detailing checks.

Sources: ACI 318-19, §25.4.2 (development), §25.5 (splices), Table 25.4.2.5 (modification factors). Wight & MacGregor, Reinforced Concrete: Mechanics and Design. Values computed for f_y = 60,000 psi; Grade 80 bars scale by 80/60 plus additional ACI provisions.

Card: pe-calc.com/cheat-sheets/rebar-development-length

Design Rainfall — NOAA Atlas 14 / IDF Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

Where design storm depths come from, which precipitation-frequency source covers your state, and how to move between depth, duration, and intensity without the classic mistakes.

The Workflow

Step	What to do
1	Open the PFDS (hdsc.nws.noaa.gov/pfds), click the site or enter lat/lon.
2	Read the depth-duration-frequency (DDF) table — depths by duration (5 min–60 day) and ARI (1–1000 yr), with 90% bounds.
3	Rational Method: read depth at D = t_c, convert i = P/D. NRCS methods: take the 24-hr depth and shape it with the applicable storm distribution.
4	Record station/grid, volume, and retrieval date in the drainage report — estimates change when volumes are updated.

Coverage by Source

Region	Source
Ohio Valley & mid-Atlantic/upper Southeast (NC, SC, VA, WV, TN, KY, OH, IN, IL, PA, NJ, MD, DE, DC)	Atlas 14 Vol. 2 (2004)
Deep Southeast (AL, AR, FL, GA, LA, MS)	Atlas 14 Vol. 9 (2013)
Midwest (CO, IA, KS, MI, MN, MO, ND, NE, OK, SD, WI)	Atlas 14 Vol. 8 (2013)
Northeast (New England, NY)	Atlas 14 Vol. 10 (2015)
Texas	Atlas 14 Vol. 11 (2018)
Semiarid Southwest (AZ, NV, NM, UT), California, Alaska, islands	Atlas 14 Vols. 1, 6, 7, 3–5
WA, OR, ID, MT, WY	NOAA Atlas 2 (1973) + state studies

NOAA Atlas 15 (in development) will supersede all of the above with nationwide, climate-informed estimates — watch the governing agency for adoption.

Depth ↔ Intensity

i (in/hr) = P (in) / D (hr)

Duration	Example depth P	Intensity i
5 min (0.083 hr)	0.45 in	5.4 in/hr
15 min (0.25 hr)	0.90 in	3.6 in/hr
1 hr	1.8 in	1.8 in/hr
24 hr	4.8 in	0.2 in/hr

Illustrative depths for a single hypothetical return period, to show the duration effect — pull the actual values for your site from PFDS. The 24-hr average intensity is useless for Rational Method work; the short-duration cell is where storm sewers live.

Three classic pitfalls. (1) Using the 24-hr depth ÷ 24 as a Rational intensity — understates i by 5–15×. (2) Grabbing depths from an old drainage manual table instead of current PFDS — many manuals predate Atlas 14 updates. (3) Pairing an Atlas 14 depth with the wrong storm distribution — several states have moved from legacy Type II/III to NRCS Atlas-14-based regional curves. The storm distribution card covers which applies where.

Sources: NOAA Atlas 14, Precipitation-Frequency Atlas of the United States, Vols. 1–11, NWS Hydrometeorological Design Studies Center; PFDS at hdsc.nws.noaa.gov. NRCS National Engineering Handbook Part 630 Ch. 4 (storm distributions). NOAA Atlas 2 (1973) for the Northwest gap.

Card: pe-calc.com/cheat-sheets/design-rainfall-atlas14

Water Properties by Temperature — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

Density, specific weight, dynamic and kinematic viscosity, and vapor pressure of fresh water at atmospheric pressure. These four properties feed nearly every hydraulic calculation — Reynolds number, friction loss, settling velocity, and pump NPSH.

The Four Values You Actually Look Up

Property	SI (20°C)	US (60°F)	Where it is used
Density ρ	998.2 kg/m³	1.938 slug/ft³	Momentum, drag, Reynolds number
Specific weight γ	9.789 kN/m³	62.37 lb/ft³	Hydrostatic pressure, buoyancy, uplift
Kinematic viscosity ν	1.003×10⁻⁶ m²/s	1.21×10⁻⁵ ft²/s	Reynolds number, friction factor
Vapor pressure p_v	2.34 kPa abs	0.256 psia	Cavitation, NPSH available

The engineer's shortcut. At ordinary ambient temperatures, γ = 62.4 lb/ft³ (9.81 kN/m³) and ν = 1.0×10⁻⁶ m²/s are accurate to better than 1–2% and are what most design manuals assume. Reach for the full table when you are checking cavitation, cold-weather friction loss, or a settling / thickening process — those are the three places temperature genuinely bites.

SI Units — Fresh Water, 0 to 100°C

T (°C)	ρ (kg/m³)	γ (kN/m³)	μ (×10⁻³ Pa·s)	ν (×10⁻⁶ m²/s)	p_v (kPa abs)
0	999.8	9.806	1.781	1.785	0.61
5	1000.0	9.807	1.518	1.519	0.87
10	999.7	9.804	1.307	1.306	1.23
15	999.1	9.798	1.139	1.139	1.70
20	998.2	9.789	1.002	1.003	2.34
25	997.0	9.777	0.890	0.893	3.17
30	995.7	9.764	0.798	0.800	4.24
40	992.2	9.730	0.653	0.658	7.38
50	988.0	9.689	0.547	0.553	12.33
60	983.2	9.642	0.466	0.474	19.92
70	977.8	9.589	0.404	0.413	31.16
80	971.8	9.530	0.354	0.364	47.34
90	965.3	9.466	0.315	0.326	70.10
100	958.4	9.399	0.282	0.294	101.33

US Customary — Fresh Water, 32 to 212°F

T (°F)	ρ (slug/ft³)	γ (lb/ft³)	μ (×10⁻⁵ lb·s/ft²)	ν (×10⁻⁵ ft²/s)	p_v (psia)
32	1.940	62.42	3.732	1.924	0.0885
40	1.940	62.43	3.228	1.664	0.122
50	1.940	62.41	2.730	1.407	0.178
60	1.938	62.37	2.344	1.210	0.256
70	1.936	62.30	2.034	1.051	0.363
80	1.934	62.22	1.791	0.926	0.507
100	1.927	62.00	1.423	0.739	0.949
120	1.918	61.71	1.164	0.607	1.69
140	1.908	61.38	0.974	0.511	2.89
160	1.896	61.00	0.832	0.439	4.74
180	1.883	60.58	0.721	0.383	7.51
200	1.869	60.12	0.634	0.339	11.52
212	1.860	59.83	0.589	0.317	14.70

Relationships

γ = ρg · ν = μ/ρ · Re = VL/ν

Use g = 9.807 m/s² (32.17 ft/s²). Note the unit trap in US customary work: density is in slug/ft³, not lb/ft³. The familiar 62.4 lb/ft³ is specific weight γ, not density ρ. Mixing the two is the single most common error in US-unit fluid calculations — they differ by the factor g = 32.17.

Where Temperature Actually Changes the Answer

Calculation	Property	Sensitivity
Hydrostatic pressure, uplift, buoyancy	γ	Negligible — under 1% across 0–30°C
Reynolds number, friction factor	ν	Strong — ν nearly halves from 5°C to 30°C
Pump NPSH available / cavitation	p_v	Strong — p_v rises ~5× from 5°C to 30°C
Particle settling (Stokes range)	μ	Strong — settling velocity scales as 1/μ

Cold water is not the conservative assumption. Cold water is more viscous, so it gives a lower Reynolds number and higher friction loss — conservative for pipe sizing. But warm water has a much higher vapor pressure, so it is the governing case for cavitation and NPSH. Check pump suction at the warmest expected water temperature and pipe friction at the coldest.

Sources: White, F.M., Fluid Mechanics, Table A.1 (water properties). Munson, Young & Okiishi, Fundamentals of Fluid Mechanics, Tables B.1 and B.2. Vapor pressures follow the IAPWS-IF97 / NIST steam-table formulation. Values are for fresh water at atmospheric pressure; dissolved solids, and salinity in particular, shift density and vapor pressure and warrant source-specific data.

Card: pe-calc.com/cheat-sheets/water-properties

Dam Breach Parameters — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

The four regression equation sets used in practice to estimate breach width, side slope, and failure time for embankment dams. All equations below are metric — V in m³, h and B in m, g = 9.81 m/s². Convert before and after; these regressions are not unit-agnostic.

Notation

Symbol	Meaning	Units
B̄	Average breach width	m
V_w	Reservoir volume at time of failure	m³
V_out	Volume of water discharged through breach	m³
h_b	Breach height (invert to crest)	m
h_w	Depth of water above breach invert at failure	m
V_er	Volume of embankment material eroded	m³
K_o	Failure-mode factor (overtopping vs piping)	—
t_f	Breach formation (failure) time	s or hr

The Four Equation Sets

Froehlich (2008) — current default

B̄ = 0.27 · K_o · V_w^0.32 · h_b^0.04
t_f = 63.2 · √[ V_w / (g · h_b²) ] (seconds)

K_o = 1.3 overtopping, 1.0 piping. Side slope 1.0H:1V overtopping, 0.7H:1V piping. Based on 74 case histories — the largest dataset of the four, and the reason this set is the usual starting point.

Froehlich (1995)

B̄ = 0.1803 · K_o · V_w^0.32 · h_b^0.19
t_f = 0.00254 · V_w^0.53 · h_b^−0.90 (hours)

K_o = 1.4 overtopping, 1.0 piping. Side slope 1.4H:1V overtopping, 0.9H:1V piping. Superseded by the 2008 set but still cited and still offered in most software.

MacDonald & Langridge-Monopolis (1984)

V_er = 0.0261 · (V_out · h_w)^0.769 (earthfill)
t_f = 0.0179 · V_er^0.364 (hours)

Predicts eroded volume rather than width directly — breach geometry is then back-figured from the embankment section, assuming a trapezoid with 0.5H:1V side slopes. The distinct approach is exactly why it is a useful independent check.

Von Thun & Gillette (1990)

B̄ = 2.5 · h_w + C_b
t_f = 0.02 · h_w + 0.25 (erosion-resistant, hours)
t_f = 0.015 · h_w (easily erodible, hours)

The only set that keys failure time to erodibility rather than volume alone, which makes it the useful bracket when embankment materials are known.

Failure-Mode Factor K_o and Side Slopes

Method	K_o overtopping	K_o piping	Side slope z (H:1V)
Froehlich (2008)	1.3	1.0	1.0 OT / 0.7 piping
Froehlich (1995)	1.4	1.0	1.4 OT / 0.9 piping
MacDonald & L-M	—	—	0.5 (assumed)
Von Thun & Gillette	—	—	1.0 typical

Von Thun & Gillette Offset C_b

Reservoir volume V_w (m³)	V_w (ac-ft)	C_b (m)	C_b (ft)
< 1.23×10⁶	< 1,000	6.1	20
1.23×10⁶ – 6.17×10⁶	1,000 – 5,000	18.3	60
6.17×10⁶ – 1.23×10⁷	5,000 – 10,000	42.7	140
> 1.23×10⁷	> 10,000	54.9	180

Worked Comparison — All Four Methods

Small high-hazard embankment, overtopping failure. V_w = 500 ac-ft (616,800 m³), h_b = h_w = 30 ft (9.14 m), earthfill.

Method	B̄ (m)	B̄ (ft)	t_f (hr)
Froehlich (2008)	27.3	90	0.48
Froehlich (1995)	27.4	90	0.41
Von Thun & Gillette	29.0	95	0.43 resistant / 0.14 erodible
MacDonald & L-M	V_er = 4,060 m³	—	0.37

Breach widths cluster within about 6% here, but failure time spans 0.14 to 0.48 hours — a factor of 3.4. That spread is the whole point of running more than one method.

Failure time drives peak outflow, not breach width. Peak breach discharge is far more sensitive to t_f than to B̄, so the parameter with the widest prediction band is also the one the answer hinges on. Froehlich (2008) reports roughly ±⅓ uncertainty on width and closer to a factor of two on time. Run the range, not a single deterministic number, and state the range in the report.

Practice Notes

Issue	Guidance
Which method to lead with	Froehlich (2008) — largest dataset, HEC-RAS default for embankments
Concrete gravity / arch dams	These regressions do not apply. Use monolith-loss assumptions per FERC / USACE guidance
Very small dams	Case-history datasets thin out below about 6 m height; treat results as indicative
Regulatory submittals	Most state programs expect a sensitivity range across at least two methods
V_w definition	Volume at the moment of failure, not normal pool — for an overtopping case that is the routed peak

Sources: Froehlich, D.C. (2008), "Embankment Dam Breach Parameters and Their Uncertainties," J. Hydraulic Engineering 134(12), 1708–1721. Froehlich, D.C. (1995), "Embankment Dam Breach Parameters Revisited," ASCE Water Resources Engineering, 887–891. MacDonald, T.C. & Langridge-Monopolis, J. (1984), "Breaching Characteristics of Dam Failures," J. Hydraulic Engineering 110(5), 567–586. Von Thun, J.L. & Gillette, D.R. (1990), "Guidance on Breach Parameters," USBR. See also USACE HEC-RAS Hydraulic Reference Manual (dam breach chapter) and FERC Engineering Guidelines Chapter 2. Comparison values above were computed directly from the equations as written.

Card: pe-calc.com/cheat-sheets/dam-breach-parameters

Bridge Scour Equations — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

The FHWA HEC-18 equation set for pier and contraction scour: CSU and Froehlich for local pier scour, Laursen for contraction scour, with the K-factor tables. Equations below are dimensionally consistent — use one unit system throughout, and note that only the clear-water contraction coefficient K_u changes between SI and US customary.

Total Scour Is a Sum of Three Components

y_s,total = y_degradation + y_contraction + y_local

Component	What it is	Method
Long-term degradation	Channel-wide bed lowering over years, independent of the bridge	Geomorphic assessment, HEC-20
Contraction scour	Bed lowering across the whole opening from flow constriction	Laursen (live-bed or clear-water)
Local scour	The hole at an individual pier or abutment from vortex action	CSU / Froehlich (piers); Froehlich / HIRE (abutments)

They are computed independently and added. HEC-18 expects evaluation at both a design flood and a check flood — commonly the 100-year and 500-year events.

Live-Bed vs Clear-Water — Decide First

Condition	Criterion	Behaviour
Live-bed scour	V₁ > V_c	Sediment continuously resupplied; depth oscillates about an equilibrium
Clear-water scour	V₁ < V_c	No resupply; hole deepens asymptotically to a maximum

This is a fork, not a maximum. Compare approach velocity V₁ against the critical velocity V_c for the bed D₅₀, then use the matching contraction equation. Computing both and taking the larger is a common and wrong shortcut — clear-water often returns the bigger number in a case where live-bed physically governs.

Local Pier Scour

CSU equation (HEC-18 primary)

y_s = 2.0 · y₁ · K₁ · K₂ · K₃ · (a / y₁)^0.65 · Fr₁^0.43

y₁ = approach flow depth, a = pier width, Fr₁ = V₁/√(g·y₁). Upper limits: y_s ≤ 2.4·a for Fr₁ ≤ 0.8, and y_s ≤ 3.0·a for Fr₁ > 0.8.

Froehlich equation (design form)

y_s = 0.32 · φ · (a′)^0.62 · y₁^0.47 · Fr₁^0.22 · D₅₀^−0.09 + a

φ = 1.3 square nose, 1.0 round nose, 0.7 sharp nose. a′ = pier width projected normal to the flow. The trailing + a is a deliberate safety addition, which is why Froehlich is often used as a check rather than the primary. Unlike CSU, this form is sensitive to D₅₀.

Pier Correction Factors

K₁ — pier nose shape

Nose shape	K₁
Square nose	1.1
Round nose	1.0
Circular cylinder	1.0
Group of cylinders	1.0
Sharp (triangular) nose	0.9

K₂ — angle of attack

K₂ = ( cos θ + (L / a) · sin θ )^0.65

Angle θ	L/a = 4	L/a = 8	L/a = 12
0°	1.0	1.0	1.0
15°	1.5	2.0	2.5
30°	2.0	2.75	3.5
45°	2.3	3.3	4.3
90°	2.5	3.9	5.0

K₂ dominates everything else when flow is skewed. A 30° skew on a long pier multiplies scour by 3.5 — if K₂ > 1, nose shape stops mattering and HEC-18 directs you to use K₁ = 1.0.

K₃ — bed condition

Bed condition	Dune height H	K₃
Clear-water scour	—	1.1
Plane bed and antidune flow	—	1.1
Small dunes	0.6–3 m (2–10 ft)	1.1
Medium dunes	3–9 m (10–30 ft)	1.1–1.2
Large dunes	≥ 9 m (30 ft)	1.3

Contraction Scour — Laursen

Live-bed

y₂ / y₁ = (Q₂ / Q₁)^6/7 · (W₁ / W₂)^k₁

V_* / ω	Mode of bed-material transport	k₁
< 0.50	Mostly contact bed-material discharge	0.59
0.50 – 2.0	Some suspended bed-material discharge	0.64
> 2.0	Mostly suspended bed-material discharge	0.69

V_* = √(g·y₁·S₁) is shear velocity and ω is the fall velocity of the bed D₅₀ — see the settling velocity card.

Clear-water

y₂ = [ K_u · Q² / ( D_m^2/3 · W² ) ]^3/7

Term	SI	US customary
K_u	0.025	0.0077
D_m (effective diameter)	1.25 · D₅₀	1.25 · D₅₀

In both forms, contraction scour depth is y₂ minus the existing bed depth in the contracted section, not y₂ itself.

Worked Example

Circular pier, a = 4 ft. Approach y₁ = 10 ft, V₁ = 8 ft/s, flow aligned. Bed D₅₀ = 2 mm, plane bed (K₃ = 1.1). Opening contracts from W₁ = 400 ft to W₂ = 250 ft, all flow through the bridge, live-bed with k₁ = 0.64.

Step	Result
Fr₁ = 8 / √(32.2 × 10)	0.446
CSU pier scour = 2.0(10)(1.0)(1.0)(1.1)(0.4)^0.65(0.446)^0.43	8.57 ft
CSU limit check, Fr ≤ 0.8 → 2.4a = 9.6 ft	OK, not governed
Froehlich pier scour (φ = 1.0, a′ = 4 ft)	6.94 ft
Laursen live-bed y₂ = 12(1)^6/7(400/250)^0.64	16.21 ft
Contraction scour = 16.21 − 12	4.21 ft
Total (contraction + CSU local)	12.78 ft

CSU returns 8.57 ft against Froehlich's 6.94 ft — about 23% apart, which is typical. CSU is the HEC-18 primary; Froehlich is the check. Long-term degradation would be added on top from a geomorphic assessment.

Scour equations are conservative by construction. They were regressed largely on flume data at equilibrium, and field-measured scour is frequently shallower than predicted. That conservatism is deliberate — the failure mode is a bridge in a river. Do not "calibrate it out" without measured site data and a very clear rationale in the record.

Practice Notes

Issue	Guidance
Which pier equation governs	CSU is the HEC-18 primary; run Froehlich as an independent check
Skewed flow	K₂ swamps every other factor. If K₂ > 1, set K₁ = 1.0
Debris	Effective pier width increases — HEC-18 gives a debris-width procedure; do not ignore it on small streams
Design vs check flood	Evaluate both; the check flood (often 500-yr) may govern the foundation
Countermeasures	Riprap sizing at piers follows HEC-23, not the channel riprap methods
Abutments	Not covered here — use Froehlich or HIRE per HEC-18, with the NCHRP 24-20 approach where applicable

Sources: Arneson, L.A., Zevenbergen, L.W., Lagasse, P.F. & Clopper, P.E. (2012), Evaluating Scour at Bridges, 5th ed., FHWA HEC-18 (FHWA-HIF-12-003) — CSU equation, K-factor tables, Laursen contraction relations. Froehlich, D.C. (1988), "Analysis of Onsite Measurements of Scour at Piers," ASCE Hydraulic Engineering. Laursen, E.M. (1960), "Scour at Bridge Crossings," J. Hydraulics Division 86(HY2), and (1963), "An Analysis of Relief Bridge Scour," 89(HY3). See also FHWA HEC-20 (stream stability) and HEC-23 (countermeasures). Worked-example values above were computed directly from the equations as written.

Card: pe-calc.com/cheat-sheets/bridge-scour-equations

Unified Soil Classification (USCS) — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

ASTM D2487 group symbols and names, the decision rules that produce them, and the AASHTO cross-walk. Everything below comes from two lab results: a sieve/hydrometer curve and the Atterberg limits. The symbol is not a soil property — it is a shorthand for a set of expected behaviors, listed at the bottom of the card.

The Two-Letter Code

First letter — dominant fraction	Second letter — modifier
G gravel · S sand · M silt · C clay · O organic · Pt peat	W well graded · P poorly graded · M silty fines · C clayey fines · L low plasticity (LL < 50) · H high plasticity (LL ≥ 50)

Decision Sequence

> 50% retained on No. 200 → coarse-grained → > 50% of coarse fraction retained on No. 4 → gravel, else sand
≥ 50% passing No. 200 → fine-grained → plasticity chart (LL vs. PI)

Sieve sizes that matter: No. 4 = 4.75 mm, No. 40 = 0.425 mm, No. 200 = 0.075 mm. Percent fines is percent passing the No. 200. Particles larger than 3 in. are excluded from the classification and reported separately as cobbles and boulders.

Coarse-Grained Soils (> 50% retained on No. 200)

Symbol	Group name	Criteria
GW	Well-graded gravel	< 5% fines; C_u ≥ 4 and 1 ≤ C_c ≤ 3
GP	Poorly graded gravel	< 5% fines; fails either C_u or C_c
GW-GM GW-GC	Well-graded gravel with silt / with clay	5–12% fines; meets GW gradation; fines plot below / above A-line
GP-GM GP-GC	Poorly graded gravel with silt / with clay	5–12% fines; fails GW gradation
GM	Silty gravel	> 12% fines; fines are ML or MH
GC	Clayey gravel	> 12% fines; fines are CL or CH
GC-GM	Silty, clayey gravel	> 12% fines; fines are CL-ML
SW	Well-graded sand	< 5% fines; C_u ≥ 6 and 1 ≤ C_c ≤ 3
SP	Poorly graded sand	< 5% fines; fails either C_u or C_c
SW-SM SW-SC	Well-graded sand with silt / with clay	5–12% fines; meets SW gradation
SP-SM SP-SC	Poorly graded sand with silt / with clay	5–12% fines; fails SW gradation
SM	Silty sand	> 12% fines; fines are ML or MH
SC	Clayey sand	> 12% fines; fines are CL or CH
SC-SM	Silty, clayey sand	> 12% fines; fines are CL-ML

C_u = D₆₀ / D₁₀ C_c = D₃₀² / (D₁₀ · D₆₀)

Fine-Grained Soils (≥ 50% passing No. 200)

Symbol	Group name	Criteria (on the −No. 40 fraction)
CL	Lean clay	LL < 50; PI > 7 and on or above A-line
ML	Silt	LL < 50; PI < 4 or below A-line
CL-ML	Silty clay	LL < 50; 4 ≤ PI ≤ 7 and on or above A-line
OL	Organic clay / organic silt	LL (oven-dried) / LL (not dried) < 0.75
CH	Fat clay	LL ≥ 50; on or above A-line
MH	Elastic silt	LL ≥ 50; below A-line
OH	Organic clay / organic silt	LL ≥ 50; oven-dried LL ratio < 0.75
Pt	Peat	Primarily organic matter, dark, organic odor — visual/manual

A-line: PI = 0.73 (LL − 20) U-line: PI = 0.9 (LL − 8)

Plot the point (LL, PI). Above the A-line is clay, below is silt; LL = 50 splits L from H. A point above the U-line is an upper-bound violation — re-run the limits before you believe it. PI = LL − PL.

Group-Name Modifiers

Condition	Add to the group name
Coarse-grained soil, 15% or more of the other coarse fraction	“with gravel” / “with sand”
Fine-grained soil, 15–29% retained on No. 200	“with sand” or “with gravel” (whichever is larger)
Fine-grained soil, ≥ 30% retained on No. 200	“sandy” or “gravelly” prefix
Any soil with cobbles or boulders present	“with cobbles” / “with boulders”

AASHTO M145 Cross-Walk (approximate)

AASHTO	Description	Subgrade rating	Usual USCS equivalents
A-1-a	Stone fragments, gravel	Excellent	GW, GP, GM
A-1-b	Coarse sand	Excellent	SW, SP, GM, SM
A-3	Fine sand, nonplastic	Excellent to good	SP
A-2-4 A-2-5	Silty gravel & sand	Excellent to good	GM, SM
A-2-6 A-2-7	Clayey gravel & sand	Good to fair	GC, SC
A-4	Silt, LL ≤ 40, PI ≤ 10	Fair to poor	ML, OL
A-5	Elastic silt, LL > 40, PI ≤ 10	Fair to poor	MH, OH
A-6	Clay, LL ≤ 40, PI > 10	Poor	CL
A-7-5 A-7-6	Clay, LL > 40, PI > 10	Poor	CH, OH

GI = (F − 35)[0.2 + 0.005(LL − 40)] + 0.01(F − 15)(PI − 10)

Group index, with F = percent passing No. 200. Report as a whole number in parentheses after the group, e.g. A-6(9); a negative result is reported as 0. AASHTO splits granular from silt-clay at 35% passing the No. 200 — not 50% — which is why the cross-walk above can only ever be approximate.

Typical Engineering Characteristics by Group

USCS	γ_d,max (pcf)	φ′ (deg)	Drainage	Value as embankment / fill
GW	125–135	33–40	Excellent	Very stable; shell and drainage zones
GP	115–125	32–38	Excellent	Reasonably stable; pervious shell
GM	120–135	30–35	Fair to poor	Reasonably stable; impervious core if well compacted
GC	115–130	28–35	Poor	Fairly stable; impervious core and blanket
SW	110–130	33–38	Excellent	Very stable; shell and filter material
SP	100–120	30–36	Excellent	Reasonably stable when dense; liquefaction-prone if loose and saturated
SM	110–125	29–35	Fair to poor	Fairly stable; erosion- and piping-sensitive — filter it
SC	105–125	27–34	Poor	Fairly stable; usable core material
ML	95–120	26–32	Poor	Poor stability; frost-susceptible, highly erodible
CL	95–120	22–30	Practically impervious	Good stability; standard core material
OL	80–100	20–28	Poor	Not suitable — strip it
MH	70–95	23–30	Poor	Poor stability; high shrink/swell and compressibility
CH	75–105	17–25	Practically impervious	Fair stability; expansive — watch slope movement
OH	65–100	15–25	Practically impervious	Not suitable
Pt	—	—	—	Not suitable for any structural use

γ_d,max is standard Proctor (ASTM D698) maximum dry density; φ′ is the effective friction angle of compacted fill. Ranges are representative for preliminary work — site-specific testing governs final design.

The symbol is a starting point, not a design value. Two soils that both classify SM can differ by three orders of magnitude in permeability depending on how the fines are distributed. Classify to communicate and to bracket expectations; test to design. For embankment dams in particular, the piping and internal-erosion behavior of a SM/ML fill is governed by the gradation curve shape, not by the group symbol.

Sources: ASTM D2487, Standard Practice for Classification of Soils for Engineering Purposes (Unified Soil Classification System); ASTM D2488 (visual-manual). AASHTO M145 / ASTM D3282. Typical-property ranges after NAVFAC DM 7.01, Soil Mechanics, and USBR Design of Small Dams.

Card: pe-calc.com/cheat-sheets/unified-soil-classification

Soil Permeability (k) — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

Saturated hydraulic conductivity by soil type, the unit conversions that trip everyone up, and the test method that is actually valid at each magnitude. Permeability spans twelve orders of magnitude across ordinary soils — more than any other geotechnical parameter — so the useful question is almost never “what is k” but “what decade is k in, and how confident am I.”

Darcy's Law

v = k · i Q = k · i · A i = Δh / L
v_seepage = v / n (actual pore velocity; n = porosity)

v is the discharge velocity through the gross cross-section — it is not the speed of a water particle. Darcy's law holds for laminar flow, which covers essentially all soils finer than coarse gravel (Re < 1 based on D₁₀). In clean cobbles and rockfill it breaks down and a nonlinear Forchheimer form is needed.

Unit Conversions

From	→ cm/s	→ ft/day	→ m/day	→ in/hr	→ gpd/ft²
1 cm/s	1	2,835	864	1,417	21,200
1 ft/day	3.53×10⁻⁴	1	0.305	0.500	7.48
1 m/day	1.16×10⁻³	3.28	1	1.64	24.5
1 in/hr	7.06×10⁻⁴	2.00	0.610	1	15.0
1 µm/s	1×10⁻⁴	0.284	0.0864	0.142	2.12
1 gpd/ft²	4.72×10⁻⁵	0.134	0.0408	0.0669	1
1 darcy (water, 20°C)	≈9.6×10⁻⁴	2.7	0.83	1.4	20

Handy anchor: 1 cm/s ≈ 2,835 ft/day, and 1 in/hr ≈ 2 ft/day. NRCS soil surveys report µm/s; geotech reports report cm/s; groundwater models want ft/day; stormwater infiltration rules are written in in/hr.

Order of Magnitude & Drainage Class

k (cm/s)	Drainage	Typical soils	What it means in practice
10²–10⁰	Very good	Clean gravel, rockfill, open-graded stone	Free-draining; drain rock, chimney and blanket drains
10⁰–10⁻³	Good	Clean sands, clean sand-gravel mixtures	Drains under gravity; usable as a filter or infiltration receiver
10⁻³–10⁻⁵	Poor	Very fine sands, silts, silty/clayey sands, glacial till	Slow drainage; frost-susceptible; marginal for infiltration BMPs
10⁻⁵–10⁻⁷	Very poor	Silt, stratified clay, weathered clay fill	Effectively a barrier over construction time scales
< 10⁻⁷	Practically impervious	Homogeneous clays below the weathered zone, CCLs, GCLs	Liner and core material; 1×10⁻⁷ cm/s is the common CCL spec

Typical k by USCS Group (compacted fill)

USCS	k (cm/s)	k (ft/day)	Role in an embankment / earthwork
GW	10⁻²–10⁰	30–2,800	Pervious shell, drainage zone
GP	10⁻¹–10¹	300–28,000	Drain rock; needs a filter against migration
GM	10⁻⁶–10⁻³	0.003–3	Semi-pervious; usable core if fines are plastic
GC	10⁻⁸–10⁻⁶	3×10⁻⁵–0.003	Good core and blanket material
SW	10⁻³–10⁻¹	3–300	Shell; good filter sand
SP	10⁻³–10⁻¹	3–300	Shell; uniform — check filter compatibility both ways
SM	10⁻⁶–10⁻⁴	0.003–0.3	Semi-pervious; the classic internal-erosion problem soil
SC	10⁻⁸–10⁻⁶	3×10⁻⁵–0.003	Core material
ML	10⁻⁶–10⁻⁴	0.003–0.3	Erodible and dispersive-prone; avoid unfiltered
CL	10⁻⁹–10⁻⁷	3×10⁻⁶–3×10⁻⁴	Standard impervious core
MH	10⁻⁸–10⁻⁶	3×10⁻⁵–0.003	Poor fill; high compressibility
CH	10⁻¹⁰–10⁻⁸	3×10⁻⁷–3×10⁻⁵	Very low k, but shrink/swell cracking can dominate the field value

Ranges are for soil compacted near standard Proctor optimum. Compacting wet of optimum can drop k by one to two orders of magnitude versus the same soil compacted dry of optimum — the placement water content is often a bigger lever on k than the material selection.

Estimating k from Gradation

Hazen: k (cm/s) = C · D₁₀² (D₁₀ in mm, C ≈ 0.4–1.2, use 1.0)
Kozeny–Carman: k ∝ e³ / (1 + e) (void-ratio dependence within one soil)

Hazen is valid only for clean, loose to medium-dense sand with C_u < 5 and D₁₀ between about 0.1 and 3.0 mm. Outside that window it is not conservative in either direction. Treat it as a sanity check on a measured value, never as a substitute for one.

Test Methods & Their Valid Range

Method	Usable k (cm/s)	Notes
Constant head, rigid wall — ASTM D2434	> 10⁻³	Coarse soils; watch sidewall leakage and turbulence at high gradients
Falling head, rigid wall	10⁻³–10⁻⁶	Fine sands and silts
Flexible wall (triaxial) — ASTM D5084	≤ 10⁻⁶	The standard for liners and cores; back-pressure saturate first
Oedometer, from consolidation — ASTM D2435	10⁻⁷–10⁻¹⁰	Indirect: k = c_v · m_v · γ_w
Pumping test (field, saturated)	> 10⁻⁵	Best mass value; averages fabric and stratification over a large volume
Slug / bail test (field)	10⁻²–10⁻⁷	Samples only the material near the well screen
Double-ring infiltrometer — ASTM D3385	> 10⁻⁵	Vadose-zone infiltration rate, not saturated k; the usual BMP test
Borehole / Guelph permeameter	10⁻³–10⁻⁶	Field-saturated k above the water table

Field mass permeability commonly comes out 10 to 1,000× higher than a lab value on an intact tube sample, because the lab specimen misses the sand seams, root holes, desiccation cracks and lift interfaces that carry most of the flow. Use lab values for the core spec and field values for the seepage estimate.

Anisotropy & Temperature

Condition	k_h / k_v
Homogeneous, isotropic (an assumption, rarely a fact)	1
Compacted embankment fill placed in lifts	4–9
Natural stratified alluvium	2–10
Interbedded sand and clay, varved clay	10–100+

k₂₀ = k_T · (μ_T / μ₂₀)

Permeability is reported at 20°C. Because k scales inversely with viscosity, cold water moves roughly 40% slower at 5°C than at 20°C — a real effect for winter infiltration performance and for lab tests run in an unheated shed. Viscosity values are on the water properties card.

NRCS Hydrologic Soil Groups (K_sat of the least transmissive layer)

HSG	K_sat (in/hr)	K_sat (µm/s)	Typical texture
A	> 5.67	> 40	Sand, loamy sand, sandy loam — deep, well drained
B	1.42–5.67	10–40	Silt loam, loam
C	0.14–1.42	1.0–10	Sandy clay loam
D	< 0.14	< 1.0	Clay loam, silty clay, clay; also any soil over shallow bedrock or a high water table

The conductivity band applies when depth to a water-impermeable layer exceeds about 40 in. and depth to the seasonal high water table exceeds about 24 in. Shallower conditions force a dual group — A/D, B/D, C/D — which reverts to D unless the site is actually drained. These groups feed straight into the curve number card.

Two places k gets used badly. (1) Infiltration BMPs. A design rate is a field-measured rate divided by a factor of safety — commonly 2, and more where a single test represents a large footprint. Below about 0.5 in/hr, an infiltration practice is usually the wrong practice, not a practice with a longer drawdown. (2) Seepage and piping. Total seepage quantity scales with k, but internal erosion risk does not — it is governed by the exit gradient and by filter compatibility. The critical gradient for heave is i_c = (G_s − 1)/(1 + e) ≈ 1.0, and design exit gradients for embankment dams are held well below that (FS of 3 or more, i.e. i_exit ≤ ~0.3).

Sources: Terzaghi, Peck & Mesri, Soil Mechanics in Engineering Practice; Casagrande & Fadum permeability chart; NAVFAC DM 7.01, Soil Mechanics; USBR Design of Small Dams; ASTM D2434, D5084, D3385; NRCS National Engineering Handbook Part 630, Chapter 7 (hydrologic soil groups). Ranges are representative for preliminary work — site-specific testing governs final design.

Card: pe-calc.com/cheat-sheets/soil-permeability

Permissible Velocity & Shear Stress — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

What a channel lining can take before it moves — stated both ways engineers are asked for it. The velocity tables are the older, simpler screen; the tractive-force tables are what actually governs. Both are here because criteria manuals still cite both.

The Tractive Force

τ_d = γ · d · S (lb/ft²; γ = 62.4 lb/ft³, d = max depth ft, S = bed slope)
Side slope: τ_s = K₁ · τ_d Bend: τ_b = K_b · τ_d
Design check: τ_permissible ≥ SF · τ_d

Use the maximum depth, not the hydraulic radius, for the bed shear in a lining check. HEC-15 recommends a safety factor of 1.0 to 1.5, toward the upper end where failure of the lining would endanger a road embankment, a dam, or a downstream structure.

Side-slope and bend factors (HEC-15)

Factor	Relation
K₁, side slope (z:1 horizontal:vertical)	0.77 for z ≤ 1.5 · 0.066z + 0.67 for 1.5 < z < 5 · 1.0 for z ≥ 5
K_b, bend (R_c = centerline radius, T = top width)	2.00 for R_c/T ≤ 2 · 2.38 − 0.206(R_c/T) + 0.0073(R_c/T)² for 2 < R_c/T < 10 · 1.05 for R_c/T ≥ 10

Bend protection extends a length L_p downstream of the bend exit, not just through the curve — the secondary circulation takes distance to decay. A common failure is a correctly sized bend lining that stops at the point of tangency.

Permissible Velocity & Tractive Force — Unlined Channels

Fortier & Scobey values for straight channels of small slope, after aging. “Clear water” vs. “water carrying colloidal silts” — sediment-laden flow permits more because the deposited fines armor the boundary.

Material	n	V, clear (ft/s)	V, colloidal silts (ft/s)	τ_p, clear (lb/ft²)	τ_p, colloidal (lb/ft²)
Fine sand, colloidal	0.020	1.50	2.50	0.075	0.15
Sandy loam, noncolloidal	0.020	1.75	2.50	0.075	0.15
Silt loam, noncolloidal	0.020	2.00	3.00	0.11	0.22
Alluvial silts, noncolloidal	0.020	2.00	3.50	0.11	0.22
Ordinary firm loam	0.020	2.50	3.50	0.15	0.22
Volcanic ash	0.020	2.50	3.50	0.15	0.22
Stiff clay, very colloidal	0.025	3.75	5.00	0.26	0.46
Alluvial silts, colloidal	0.025	3.75	5.00	0.26	0.46
Shales and hardpans	0.025	6.00	6.00	0.67	0.67
Fine gravel	0.020	2.50	5.00	0.075	0.32
Graded loam to cobbles, noncolloidal	0.030	3.75	5.00	0.38	0.66
Graded silts to cobbles, colloidal	0.030	4.00	5.50	0.43	0.80
Coarse gravel, noncolloidal	0.025	4.00	6.00	0.30	0.67
Cobbles and shingles	0.035	5.00	5.50	0.91	1.10

Grass-Lined Channels — Permissible Velocity (NRCS)

Cover	Slope (%)	Erosion-resistant soil (ft/s)	Easily eroded soil (ft/s)
Bermudagrass	0–5	8	6
	5–10	7	5
	> 10	6	4
Buffalograss, Kentucky bluegrass, smooth brome, blue grama	0–5	7	5
	5–10	6	4
	> 10	5	3
Grass-legume mixture	0–5	5	4
Grass-legume mixture	5–10	4	3
Lespedeza sericea, weeping lovegrass, kudzu, alfalfa, crabgrass	0–5	3.5	2.5
Annuals used as temporary cover	0–5	3.5	2.5

Grass-legume mixtures and the bunch/annual covers are not recommended on slopes steeper than 10% and 5% respectively. All values assume a uniform, well-established stand — the design condition for the first storm after seeding is bare soil plus whatever temporary lining you specified.

Permissible Shear Stress by Lining (HEC-15)

Lining	τ_p (lb/ft²)	Notes
Temporary / rolled erosion control products
Woven paper net	0.15	Very short service life
Jute net	0.45
Fiberglass roving, single	0.60
Fiberglass roving, double	0.85
Straw with net	1.45	The common construction-phase lining
Curled wood mat	1.55
Synthetic mat	2.00
Vegetative, by retardance class
Class A — very high retardance	3.70	Excellent stand, tall (~30 in.): weeping lovegrass, yellow bluestem
Class B — high	2.10	Good stand mowed 12–24 in.: smooth brome, Bermudagrass
Class C — moderate	1.00	Good stand mowed ~6 in.; grass-legume mixture
Class D — low	0.60	Good stand mowed ~2.5 in.; buffalograss
Class E — very low	0.35	Cut to 1.5 in.; burned or sparse stubble
Gravel & riprap
Gravel, D₅₀ = 1 in.	0.33	All four follow τ_p ≈ 4 · D₅₀(ft)
Gravel, D₅₀ = 2 in.	0.67
Rock riprap, D₅₀ = 6 in.	2.00
Rock riprap, D₅₀ = 12 in.	4.00

Required D₅₀ (ft) ≈ SF · τ_d / 4 D₅₀ (in) ≈ 3 · SF · τ_d

Retardance class is not a species — it is a species plus a stand condition plus a mowing height, and it changes seasonally. Design the lining for the retardance class that will exist at the worst time of year, and check capacity (Manning's n) for the class that will exist at the best time of year; those are two different classes and two different checks.

Rigid Linings

Cast-in-place concrete, grouted riprap, soil cement and articulated block are not shear-limited within the range of ordinary channel design. They fail three other ways, and those are what the design has to address:

Failure mode	What to detail
Undermining at the downstream terminus	Cutoff wall / toe-down keyed below the expected scour depth
Uplift from groundwater or from flow beneath the slab	Weep holes, underdrain, filter or geotextile bedding
Loss of subgrade support through joints and cracks	Joint sealing, reinforcement, and a graded filter under the lining

Worked check. A trapezoidal ditch runs 1.5 ft deep on a 3.0% grade, 2:1 side slopes, straight reach. τ_d = 62.4 × 1.5 × 0.03 = 2.81 lb/ft². Side slope: K₁ = 0.066(2) + 0.67 = 0.80, so τ_s = 2.25 lb/ft². Vegetated Class C (τ_p = 1.00) fails. Straw-with-net (1.45) fails. With SF = 1.0 the required riprap is D₅₀ = 2.81/4 = 0.70 ft ≈ 8.4 in., so an available 9 in. class works — and if this reach were in a bend with R_c/T = 3, K_b = 1.83 would push τ to 5.1 lb/ft² and D₅₀ to 15 in.

Sources: FHWA HEC-15, Design of Roadside Channels with Flexible Linings (permissible shear by lining, K₁ and K_b factors); Fortier & Scobey (1926) as tabulated in Chow, Open-Channel Hydraulics, Table 7-3; NRCS/SCS grass-lined channel permissible velocities. Manufacturer-tested values for a specific RECP or TRM supersede the generic product-class values above.

Card: pe-calc.com/cheat-sheets/permissible-velocity-shear

Pipe Sizes & Actual Inside Diameter — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

The number on the plan sheet is a name, not a diameter. Every hydraulic calculation on this site — Manning's, Darcy-Weisbach, Hazen-Williams, full-flow capacity — wants the actual inside diameter, which depends on material, schedule, pressure class and lining. This card is the translation table.

Steel & IPS-Dimensioned Pipe (ASME B36.10)

PVC and CPVC Schedule 40 and 80 share these same outside diameters and wall thicknesses. OD is fixed by the nominal size; the schedule changes only the wall.

NPS	OD (in)	Sch 40 wall	Sch 40 ID	Sch 80 wall	Sch 80 ID
½	0.840	0.109	0.622	0.147	0.546
¾	1.050	0.113	0.824	0.154	0.742
1	1.315	0.133	1.049	0.179	0.957
1¼	1.660	0.140	1.380	0.191	1.278
1½	1.900	0.145	1.610	0.200	1.500
2	2.375	0.154	2.067	0.218	1.939
2½	2.875	0.203	2.469	0.276	2.323
3	3.500	0.216	3.068	0.300	2.900
4	4.500	0.237	4.026	0.337	3.826
6	6.625	0.280	6.065	0.432	5.761
8	8.625	0.322	7.981	0.500	7.625
10	10.750	0.365	10.020	0.594	9.562
12	12.750	0.406	12.000	0.688	11.374
14	14.000	0.438	13.124	0.750	12.500
16	16.000	0.500	15.000	0.844	14.312
18	18.000	0.562	16.876	0.938	16.124
20	20.000	0.594	18.812	1.031	17.938
24	24.000	0.688	22.624	1.219	21.562

Above NPS 12 the nominal size is the outside diameter. Below it, OD exceeds the nominal — and 12 in. Sch 40 is the one happy coincidence where the ID comes out at exactly 12.000 in.

SDR / DR Pipe — the Ratio Does the Work

SDR = OD / t_min ID = OD × (1 − 2 / SDR)
Pressure rating = 2 · HDS / (SDR − 1)

SDR / DR	ID factor (× OD)	PVC, HDS 2,000 psi (psi @ 73°F)	HDPE PE4710, HDS 1,000 psi (psi @ 73°F)
51	0.9608	80	—
41	0.9512	100	—
32.5	0.9385	125	63
26	0.9231	160	80
25	0.9200	165	—
21	0.9048	200	100
18	0.8889	235	—
17	0.8824	250	125
14	0.8571	305	—
13.5	0.8519	315	160
11	0.8182	—	200
9	0.7778	—	250
7.3	0.7260	—	320

The SDR 41 / 32.5 / 26 / 21 / 17 / 13.5 series is ASTM D2241 PVC on IPS outside diameters. The DR 51 / 41 / 32.5 / 25 / 21 / 18 / 14 series is AWWA C900 PVC on ductile-iron outside diameters. HDPE per ASTM F714 / AWWA C906 comes in both IPS and DIPS sizing — specify which. HDPE pressure ratings are for water at 73°F and must be derated for temperature and for surge.

Worked example — how much bore a class change costs

A “6-inch” IPS PVC main, OD = 6.625 in.: SDR 41 → ID 6.30 in. · SDR 26 → 6.12 · SDR 21 → 6.00 · SDR 17 → 5.85. That is a 14% spread in flow area between the thinnest and thickest class, and roughly a 20% spread in Hazen-Williams head loss at a fixed flow — enough to change a pump selection.

Ductile Iron Pipe (AWWA C151) — Outside Diameters

Nominal	3	4	6	8	10	12	14	16	18	20	24	30	36
OD (in)	3.96	4.80	6.90	9.05	11.10	13.20	15.30	17.40	19.50	21.60	25.80	32.00	38.30

Wall thickness varies by pressure class (PC 150 through PC 350); 4 in. through 12 in. carry a 0.25 in. minimum wall in every class. Standard cement-mortar lining (AWWA C104) takes roughly another ⅛ in. off the diameter, double-thickness lining about ¼ in. Take the bare ID as OD − 2t and subtract the lining, or read the manufacturer's table — for hydraulics, the lined ID is the one that counts. DIP is normally analyzed with Hazen-Williams C = 140 new, 130 design, dropping with age and tuberculation.

Reinforced Concrete Pipe (ASTM C76)

Standard inside diameters (in)	12, 15, 18, 21, 24, 27, 30, 33, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108
Wall A	t = D/12 (in., D in in.) — thinnest
Wall B	t = D/12 + 1 — the usual default
Wall C	t = D/12 + 1.75 — heaviest

Class	D-load at 0.01 in. crack (lb/ft/ft of dia.)	Ultimate D-load	Typical use
I	800	1,200	Large diameter only (60 in. and up), shallow cover
II	1,000	1,500	Light cover, no traffic
III	1,350	2,000	The common storm-drain default
IV	2,000	3,000	Deep fill or heavy traffic
V	3,000	3,750	Very deep fill, industrial loading

D-load is a three-edge-bearing test result in pounds per linear foot per foot of inside diameter. Required D-load = (earth load + live load) × factor of safety ÷ bedding factor ÷ D — so the bedding class specified on the detail is doing as much work as the pipe class.

Corrugated Pipe

Product	Sizes	Manning's n	Notes
CMP, 2⅔ × ½ corrugation (AASHTO M36)	12–120 in.	0.024	Annular; helical in small diameters runs lower, ~0.012–0.022
CMP, 3 × 1 corrugation	36–144 in.	0.027	Larger diameters and structural plate
CMP, 5 × 1 corrugation	48–144 in.	0.025–0.026
Corrugated HDPE, dual wall (AASHTO M294)	12–60 in.	0.012	Smooth interior liner; n applies to the liner, not the corrugation
Corrugated HDPE, single wall	3–24 in.	0.020–0.025	Drainage / underdrain, not a storm main

CMP gauge is the sheet thickness: 16 ga = 0.064 in., 14 ga = 0.079, 12 ga = 0.109, 10 ga = 0.138, 8 ga = 0.168. For a corrugated pipe the nominal diameter is the inside diameter measured across the corrugation crests, and it is what you use for hydraulics.

Flow Area by Actual Inside Diameter

A (ft²) = πD²/4 = 0.005454 · D² (D in inches)

D (in)	4	6	8	10	12	15	18	24	30	36	42	48	60	72
A (ft²)	0.087	0.196	0.349	0.545	0.785	1.227	1.767	3.142	4.909	7.069	9.621	12.57	19.63	28.27

Two OD families, and they do not mix. IPS (iron pipe size, the steel ODs at the top of this card) and DIPS/CIOD (the ductile-iron ODs) are different pipes with the same names. AWWA C900 PVC and DIPS HDPE are built to the ductile-iron OD so they take ductile-iron fittings, restraints and tapping sleeves; ASTM D2241 PVC and IPS HDPE are not. Call out the OD family on the plans and in the specification, not just the nominal size and the class — the mismatch is usually discovered in the trench.

Sources: ASME B36.10M (welded and seamless wrought steel pipe); ASTM D1785 and D2241 (PVC); AWWA C900 (PVC pressure pipe), C905, C906 and ASTM F714 (PE); AWWA C151/A21.51 and C104 (ductile iron and cement-mortar lining); ASTM C76 (RCP); AASHTO M36 (CMP) and M294 (corrugated PE). Pressure ratings are for water at 73°F before surge allowance and temperature derating; confirm against the current standard and the manufacturer's submittal.

Card: pe-calc.com/cheat-sheets/pipe-sizes-dimensions

Rebar Sizes & Areas — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

ASTM A615/A706 deformed bar properties, the two tables you actually reach for on a detail sheet (total area for n bars, and area per foot of width by spacing), plus ACI 318 hook geometry and cover. Bar number = nominal diameter in eighths of an inch, exactly through #8.

Bar Properties (ASTM A615 / A706, inch-pound)

Bar	Diameter d_b (in)	Area A_b (in²)	Weight (lb/ft)	Perimeter (in)	Soft metric	Metric area (mm²)
#3	0.375	0.11	0.376	1.178	#10	71
#4	0.500	0.20	0.668	1.571	#13	129
#5	0.625	0.31	1.043	1.963	#16	199
#6	0.750	0.44	1.502	2.356	#19	284
#7	0.875	0.60	2.044	2.749	#22	387
#8	1.000	0.79	2.670	3.142	#25	510
#9	1.128	1.00	3.400	3.544	#29	645
#10	1.270	1.27	4.303	3.990	#32	819
#11	1.410	1.56	5.313	4.430	#36	1006
#14	1.693	2.25	7.650	5.319	#43	1452
#18	2.257	4.00	13.600	7.091	#57	2581

There is no #12, #13, #15, #16 or #17. Sizes #14 and #18 are mill-order items, are not generally available in short lengths, and cannot be bent in the field.

Total Area for n Bars (in²)

Bar	1	2	3	4	5	6	7	8
#3	0.11	0.22	0.33	0.44	0.55	0.66	0.77	0.88
#4	0.20	0.40	0.60	0.80	1.00	1.20	1.40	1.60
#5	0.31	0.62	0.93	1.24	1.55	1.86	2.17	2.48
#6	0.44	0.88	1.32	1.76	2.20	2.64	3.08	3.52
#7	0.60	1.20	1.80	2.40	3.00	3.60	4.20	4.80
#8	0.79	1.58	2.37	3.16	3.95	4.74	5.53	6.32
#9	1.00	2.00	3.00	4.00	5.00	6.00	7.00	8.00
#10	1.27	2.54	3.81	5.08	6.35	7.62	8.89	10.16
#11	1.56	3.12	4.68	6.24	7.80	9.36	10.92	12.48

Area per Foot of Width by Spacing (in²/ft)

Bar	3″	4″	6″	8″	9″	10″	12″	18″
#3	0.44	0.33	0.22	0.165	0.147	0.132	0.110	0.073
#4	0.80	0.60	0.40	0.300	0.267	0.240	0.200	0.133
#5	1.24	0.93	0.62	0.465	0.413	0.372	0.310	0.207
#6	1.76	1.32	0.88	0.660	0.587	0.528	0.440	0.293
#7	2.40	1.80	1.20	0.900	0.800	0.720	0.600	0.400
#8	3.16	2.37	1.58	1.185	1.053	0.948	0.790	0.527
#9	4.00	3.00	2.00	1.500	1.333	1.200	1.000	0.667

A_s (in²/ft) = A_b × 12 / s s (in) = A_b × 12 / A_s,req

Grades & Material

Spec	Grades (f_y, ksi)	Notes
ASTM A615 — carbon steel	40, 60, 80, 100	The default. No weldability limits — welding requires a carbon-equivalent check per AWS D1.4.
ASTM A706 — low-alloy	60, 80, 100	Weldable, controlled chemistry, f_t/f_y ≥ 1.25 and a capped actual yield. Required in most seismic force-resisting systems.
ASTM A996 — rail/axle	40, 50, 60	Legacy; limited availability.
ASTM A1035 — low-carbon chromium	100, 120	High strength, high corrosion resistance; design provisions are limited in ACI 318.
ASTM A775 / A934 epoxy-coated	—	Coating applies a development-length factor ψ_e = 1.2 or 1.5. Handle with padded slings.

E_s = 29,000 ksi for all grades. Yield strain ε_y = f_y/E_s: 0.00138 (Gr 40), 0.00207 (Gr 60), 0.00276 (Gr 80), 0.00345 (Gr 100). Deformed bars are furnished in 20, 40 and 60 ft stock lengths.

ACI 318 Standard Hooks & Minimum Bend Diameters

Hook type	Bar sizes	Min. inside bend dia.	Straight extension
90° standard hook (development)	#3–#8	6 d_b	12 d_b
90° standard hook (development)	#9–#11	8 d_b	12 d_b
90° standard hook (development)	#14, #18	10 d_b	12 d_b
180° standard hook	#3–#18	same as above	4 d_b, ≥ 2.5 in.
90° stirrup / tie hook	#3–#5	4 d_b	6 d_b
90° stirrup / tie hook	#6–#8	6 d_b	12 d_b
135° seismic hook	#3–#8	4–6 d_b	6 d_b, ≥ 3 in.

Minimum Concrete Cover (ACI 318, cast-in-place, non-prestressed)

Exposure	Bar sizes	Cover
Cast against and permanently in contact with ground	All	3 in.
Exposed to weather or in contact with ground (formed)	#6–#18	2 in.
Exposed to weather or in contact with ground (formed)	#5 and smaller	1½ in.
Not exposed — slabs, joists, walls	#11 and smaller	¾ in.
Not exposed — slabs, joists, walls	#14, #18	1½ in.
Not exposed — beams, columns, ties, stirrups	All	1½ in.

Spacing & Minimum Steel Rules of Thumb

Rule	Requirement
Minimum clear spacing, parallel bars in a layer	Greatest of 1 in., d_b, and (4/3)·d_agg
Maximum spacing, flexural steel in slabs & walls	Lesser of 3h and 18 in.
Shrinkage & temperature steel, Gr 60	ρ = 0.0018 · A_g (0.0020 for Gr 40/50)
Minimum flexural steel, beams	A_s,min = max(3√f′_c/f_y, 200/f_y) · b_w d
Weight of a bar mat	lb = (lb/ft from the table) × total linear feet of bar; add 5–10% for laps and waste

Nominal diameter is not the ordering diameter. The values above are the nominal plain-round equivalents used for design. The deformations make the actual outside dimension larger — roughly d_b + 0.05 to 0.10 in. — which matters for clear spacing at congested joints, for coupler and sleeve fits, and for cover at the outermost bar. Detail with the nominal value, but check tight conditions with the mill's actual deformation height.

Sources: ASTM A615/A615M, A706/A706M, A996, A1035; ACI 318-19 Ch. 20 (cover), Ch. 25 (hooks, bend diameters, spacing); CRSI Manual of Standard Practice. Areas and weights are the standard nominal values; unit weight of steel taken as 490 pcf.

Card: pe-calc.com/cheat-sheets/rebar-sizes

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

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 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

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.

Card: pe-calc.com/cheat-sheets/hds5-inlet-control-coefficients

Vertical Curve K Values — AASHTO Crest & Sag Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

K is the rate of vertical curvature — the horizontal distance in feet required to produce a 1 percent change in grade. Curve length follows directly: L = K · A, where A is the algebraic difference in grades in percent. Design K is controlled by sight distance: headlight throw on sag curves, and the driver's line of sight over the crest on crest curves.

L = K · A | K = L / A | A = |g₂ − g₁| (percent, signed grades)

Crest Vertical Curves — Stopping Sight Distance

Design speed (mph)	SSD (ft)	K (design)	Notes
15	80	3	local / parking
20	115	7
25	155	12	residential collector
30	200	19
35	250	29
40	305	44	urban arterial
45	360	61
50	425	84
55	495	114	rural two-lane
60	570	151
65	645	193
70	730	247	freeway
75	820	312
80	910	384

Crest SSD basis: driver eye height h₁ = 3.50 ft, object height h₂ = 2.00 ft.

Sag Vertical Curves — Headlight Sight Distance

Design speed (mph)	SSD (ft)	K (design)
15	80	10
20	115	17
25	155	26
30	200	37
35	250	49
40	305	64
45	360	79
50	425	96
55	495	115
60	570	136
65	645	157
70	730	181
75	820	206
80	910	231

Sag basis: headlight height 2.00 ft, upward divergence of the light beam 1°.

Sag K is larger than crest K below about 55 mph, and smaller above it. Don't assume one governs. At 30 mph sag needs K = 37 against crest's 19; at 70 mph crest needs 247 against sag's 181. Always check the curve type you actually have.

Governing Sight Distance Equations

Case	Equation (US units, L and S in ft, A in percent)
Crest, S < L	L = A·S² / 2158
Crest, S > L	L = 2S − 2158 / A
Sag (headlight), S < L	L = A·S² / (400 + 3.5·S)
Sag (headlight), S > L	L = 2S − (400 + 3.5·S) / A

The 2158 constant is 100(√(2h₁) + √(2h₂))² evaluated at h₁ = 3.50 ft and h₂ = 2.00 ft. The S > L cases rarely govern on design-speed curves but do control short curves at small A.

Other Length Criteria to Check

Criterion	Requirement	When it governs
Drainage (curbed sections)	K ≤ 167	Flat sag or crest on curb-and-gutter — caps K so a 0.3% grade is reached within 50 ft of the apex and water keeps moving
Rider comfort (sag)	L = A·V² / 46.5	Sag curves with overhead lighting where headlight criterion is relaxed; V in mph
General appearance	L ≥ 100 ft	Very small A, low speed
Minimum length	L_min = 3V	AASHTO minimum; V in mph, L in ft

The drainage maximum is the one people miss. K ≤ 167 is a maximum, not a minimum — it conflicts with the sight-distance minimum at high design speeds. On a curbed 70 mph section the crest SSD value of 247 exceeds it outright. AASHTO's guidance is that the drainage cap is not absolute; where it is exceeded, provide flanking inlets or increase gutter capacity rather than shortening the curve.

High and Low Point Location

x = K · |g₁| (distance from BVC to the turning point, ft)

Elevation at any station x from the BVC: y = y_BVC + g₁x/100 + A·x² / (200·L), with grades in percent and x, L in feet. The turning point exists on the curve only when g₁ and g₂ have opposite signs; otherwise the high or low point is at one of the tangent ends.

Sources: AASHTO, A Policy on Geometric Design of Highways and Streets (Green Book), 7th ed., 2018 — crest and sag rate-of-vertical-curvature exhibits and the stopping sight distance table. Values are the rounded design K values; confirm against the edition your agency has adopted, and against any state DOT supplement, before use on a project.

Card: pe-calc.com/cheat-sheets/vertical-curve-k-values

Bearing Capacity Factors — N_c, N_q, N_γ

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

Ultimate bearing capacity of a shallow foundation resolves into three contributions — cohesion, surcharge, and the self-weight of the failure wedge — each scaled by a dimensionless factor that depends only on the friction angle φ. Two families are in common use and they are not interchangeable: Terzaghi's original 1943 factors, and the Meyerhof/Vesić "general" factors that most modern codes build on.

Terzaghi's Equations

Strip q_u = c·N_c + q·N_q + 0.5·γ·B·N_γ Square q_u = 1.3·c·N_c + q·N_q + 0.4·γ·B·N_γ Circular q_u = 1.3·c·N_c + q·N_q + 0.3·γ·B·N_γ

where q = γ·D_f is the effective surcharge at founding level, B is the footing width (or diameter), and c is cohesion. The 1.3 / 0.4 / 0.3 multipliers are Terzaghi's empirical shape allowances, already built into the equations above — do not apply separate shape factors on top of them.

Terzaghi Bearing Capacity Factors

φ (deg)	N_c	N_q	N_γ	Typical soil
0	5.70	1.00	0.00	saturated clay, undrained
5	7.34	1.64	0.14	soft silty clay
10	9.61	2.69	0.56	silt, clayey silt
15	12.86	4.45	1.52	loose silty sand
20	17.69	7.44	3.64	loose sand
25	25.13	12.72	8.34	medium sand
30	37.16	22.46	19.13	medium–dense sand
35	57.75	41.44	45.41	dense sand, gravel
40	95.66	81.27	115.31	very dense sand / gravel
45	172.28	173.28	325.34	dense angular gravel
50	347.50	415.14	1072.80	rarely justified from field data

N_c and N_q above are Terzaghi’s closed forms. The N_γ column is the Kumbhojkar (1993) numerical evaluation of Terzaghi’s wedge solution — the set used in current editions of Das, Principles of Foundation Engineering (Table 3.1). Interpolate linearly between rows only where the interval is small; N_γ grows faster than linearly above φ = 35°, so prefer the closed forms or the calculator there.

Two different tables are both published as “Terzaghi N_γ”, and they do not agree. Terzaghi’s original 1943 values were read off a graph; Kumbhojkar later evaluated the same wedge solution numerically. Bowles, Foundation Analysis and Design (Table 4-1) carries a reconstruction closer to the original graph. The two diverge sharply in low-friction soils:

φ (deg)	Kumbhojkar / Das (used here)	Bowles Table 4-1	Difference
5	0.14	0.5	+257%
10	0.56	1.2	+114%
15	1.52	2.5	+64%
20	3.64	5.0	+37%
30	19.13	19.7	+3%
40	115.31	100.4	−13%

They agree near φ = 30° and part company on either side. If a reviewer checking your work against Bowles gets a different number, this is why — say which table you used. On cohesive soils the N_γ term is usually small enough that the divergence does not govern; on loose granular soils at low φ it can.

Meyerhof / Vesić General Factors

φ (deg)	N_c	N_q	N_γ (Vesić)
0	5.14	1.00	0.00
5	6.49	1.57	0.45
10	8.35	2.47	1.22
15	10.98	3.94	2.65
20	14.83	6.40	5.39
25	20.72	10.66	10.88
30	30.14	18.40	22.40
35	46.12	33.30	48.03
40	75.31	64.20	109.41
45	133.88	134.88	271.76
50	266.89	319.07	762.89

N_c at φ = 0 is the tell. Terzaghi gives 5.70; the general solution gives 5.14 (= 2 + π). If you are checking someone's undrained clay calculation and see q_u = 5.14·c_u + q, they used Meyerhof/Vesić. Terzaghi's factors are roughly 10–15 percent higher across the board and are the less conservative choice — but Terzaghi's method carries no depth, inclination or base-tilt factors, so a full Meyerhof/Vesić check with those applied usually lands lower.

Closed-Form Expressions

Factor	Terzaghi	Meyerhof / Vesić
N_q	e^{2(3π/4 − φ/2)tanφ} / (2cos²(45° + φ/2))	e^πtanφ · tan²(45° + φ/2)
N_c	(N_q − 1)·cotφ	(N_q − 1)·cotφ
N_γ	no closed form — tabulated from the wedge solution (Kumbhojkar 1993)	2(N_q + 1)·tanφ (Vesić)

N_c is indeterminate at φ = 0 by the cotφ form; the limiting values are 5.70 (Terzaghi) and 5.14 (general). Meyerhof's own N_γ = (N_q − 1)tan(1.4φ) differs from Vesić's and runs lower at high φ — state which one you used.

Allowable Capacity and Factor of Safety

Quantity	Expression	Note
Gross allowable	q_all = q_u / FS	FS = 3 typical for shallow foundations
Net ultimate	q_u,net = q_u − q	subtract the surcharge already there
Net allowable	q_all,net = (q_u − q) / FS	the value to compare against net applied pressure

Bearing capacity is rarely what governs. For footings on sand and on stiff clay, settlement almost always controls the design before shear failure does. Size for capacity, then check settlement — and expect the settlement check to drive the footing dimension.

Sources: Terzaghi, K. (1943). Theoretical Soil Mechanics. Meyerhof, G.G. (1963). "Some recent research on the bearing capacity of foundations," Canadian Geotechnical Journal. Vesić, A.S. (1973). "Analysis of ultimate loads of shallow foundations," JSMFD ASCE. Kumbhojkar, A.S. (1993). “Numerical evaluation of Terzaghi’s N_γ,” Journal of Geotechnical Engineering ASCE 119(3) — the source of the N_γ column here. Das, B.M., Principles of Foundation Engineering, Table 3.1. Bowles, J.E., Foundation Analysis and Design, Table 4-1 — a different N_γ reconstruction. Verify φ against actual site investigation data rather than the "typical soil" column, which is orientation only.

Card: pe-calc.com/cheat-sheets/bearing-capacity-factors

Hazen-Williams Equation — Forms, Constants & Units

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

Hazen-Williams is an empirical friction-loss relation for water in pressurized pipe. Its appeal is that roughness enters as a single coefficient C that does not depend on velocity or Reynolds number, so head loss solves in closed form. Its cost is that the equation is dimensionally inhomogeneous — the lead constant changes with every unit set, and most wrong answers come from pairing the right constant with the wrong units.

Velocity Form

Units	Equation	Where
US customary	V = 1.318·C·R^0.63·S^0.54	V in ft/s, R in ft
SI	V = 0.849·C·R^0.63·S^0.54	V in m/s, R in m

R is the hydraulic radius (D/4 for a full circular pipe) and S is the slope of the energy grade line, h_f/L — dimensionless in both systems.

Head Loss Form — the One You Actually Use

Units	Equation	Q	D	h_f, L
US, cfs & ft	h_f = 4.73·L·Q^1.852 / (C^1.852·D^4.87)	cfs	ft	ft
US, gpm & in	h_f = 10.44·L·Q^1.852 / (C^1.852·D^4.8655)	gpm	in	ft
SI	h_f = 10.67·L·Q^1.852 / (C^1.852·D^4.87)	m³/s	m	m
NFPA 13 (sprinkler)	p = 4.52·Q^1.852 / (C^1.852·d^4.87)	gpm	in	psi per ft

C is the same number in every unit system. Only the lead constant changes. If a spreadsheet is producing head losses off by a factor of several hundred, the cause is almost always diameter in inches driving a constant written for feet — D^4.87 makes a 12× unit error into roughly a 300,000× error.

Flow Form (Solved for Capacity)

Units	Equation	Q	D	Gradient term
US, cfs & ft	Q = 0.432·C·D^2.63·S^0.54	cfs	ft	S = h_f/L, ft/ft
US, gpm & in	Q = 0.282·C·D^2.63·S^0.54	gpm	in	S = h_f/L, ft/ft
US, gpm & in, pressure	Q = 0.442·C·D^2.63·(Δp/L)^0.54	gpm	in	Δp/L, psi per ft
SI	Q = 0.278·C·D^2.63·S^0.54	m³/s	m	S = h_f/L, m/m

0.442 versus 0.282 — this one bites regularly. Both are published as "the" gpm-and-inches Hazen-Williams flow constant, and they differ only in what the gradient term means. 0.442 expects pressure gradient in psi per foot; 0.282 expects head slope in feet per foot. Using 0.442 with a dimensionless head slope over-predicts capacity by about 57 percent. If you inherit a spreadsheet with 0.442 in it, confirm the S column is psi/ft before you trust the output.

These follow from the velocity form via Q = VA with R = D/4, and are the exact algebraic inverses of the head loss constants above — 0.432 → 4.73, 0.282 → 10.44, 0.278 → 10.67. Rounding of the exponent 4.87 versus 4.8655 accounts for the small differences between published values; any of them is well within the empirical accuracy of the method itself.

The Exponents, and Why They Matter

Term	Exponent	Consequence
Flow, Q	1.852	Head loss is not quadratic in Q — doubling flow raises loss by 2^1.852 ≈ 3.61×, not 4×
Roughness, C	−1.852	Dropping C from 130 to 100 raises head loss by about 63 percent
Diameter, D	−4.87	Going one nominal size up is by far the cheapest way to kill head loss
Slope, S	0.54	Reciprocal of 1.852; the two are the same relation rearranged

Validity Limits

Condition	Valid range	Outside it
Fluid	water only	Use Darcy-Weisbach — C carries no viscosity term
Temperature	~40–75°F	Error grows at both extremes; hot-water and chilled systems need D-W
Flow regime	fully turbulent	Invalid for laminar or transitional flow (Re < 4000)
Diameter	2 in and larger	Small-bore service tubing is outside the calibration set
Velocity	below ~10 ft/s	Accuracy degrades at high velocity
Pressure condition	full, pressurized	Partial-flow gravity pipe is Manning's, not Hazen-Williams

Hazen-Williams is a distribution-system tool, not a general pipe-flow tool. It was calibrated on municipal water mains at ordinary temperatures and it does that job well. EPANET, WaterCAD and most modern network solvers offer it for compatibility but compute Darcy-Weisbach internally when asked. On the PE exam, if the fluid is not water at room temperature, the problem wants Darcy-Weisbach.

Sources: Williams, G.S. & Hazen, A. (1920). Hydraulic Tables. AWWA M11 (steel pipe), M22 (sizing water service lines), M32 (distribution system modeling). NFPA 13, Standard for the Installation of Sprinkler Systems — friction loss formula. Mays, L.W. (2010). Water Distribution Systems Handbook. Hwang & Houghtalen, Fundamentals of Hydraulic Engineering Systems.

Card: pe-calc.com/cheat-sheets/hazen-williams-equation

Rational Method — Q = CiA Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

The oldest and still the most-used peak-flow method in drainage design. It returns a single number — the peak discharge — and nothing else: no hydrograph, no volume, no routing. Knowing precisely what it does and does not give you is most of using it correctly.

The Equation

Units	Equation	Q	i	A
US customary	Q = C · i · A	cfs	in/hr	acres
SI	Q = C · i · A / 360	m³/s	mm/hr	hectares
SI (alternate)	Q = 0.00278 · C · i · A	m³/s	mm/hr	hectares

Why the US form needs no conversion factor. One acre-inch per hour is 43,560 ÷ 12 ÷ 3600 = 1.008 cfs. The unit conversion is within 0.8 percent of unity, so it is dropped by convention. That coincidence is the entire reason the equation is written as bare Q = CiA in US units — it is not dimensionless, it is just very nearly 1. The SI form has no such luck, hence the 1/360.

The Three Inputs

Term	What it is	Where it comes from
C	Runoff coefficient, 0 to 1 — the fraction of rainfall that becomes direct runoff	Land-cover table; see the runoff coefficient reference
i	Average rainfall intensity over a duration equal to T_c, at the design return period	NOAA Atlas 14 IDF curves for the site
A	Contributing drainage area	Delineated to the design point

The single most common error is the intensity duration. i is not the intensity of "the 25-year storm" — it is the intensity at a duration equal to the time of concentration. Use a 24-hour depth converted to an hourly rate and you will badly under-predict the peak. Compute T_c first, then read the IDF curve at that duration.

Composite C for Mixed Land Cover

C_composite = Σ(C_j · A_j) / ΣA_j

Area-weight the individual coefficients. Do not average them unweighted, and do not apply a single "residential" C to a site that is 40 percent pavement — the weighted value is usually meaningfully higher than the eyeball estimate.

Frequency Adjustment Factor C_f

Published C values are calibrated for storms up to roughly the 10-year event. For rarer events the soil saturates and a larger fraction of rainfall runs off, so C is adjusted upward:

Return period	C_f	Applied as
2 to 10 year	1.0	C_adjusted = C · C_f
25 year	1.1
50 year	1.2
100 year	1.25

HEC-22 publishes C_f but FHWA does not endorse it. The adjustment is widely reproduced in state DOT manuals and widely used, but it is an empirical patch rather than a derived correction. Use the value your governing manual specifies; where none is specified, say in the calculations which set you applied.

Cap the product at 1.0. C · C_f greater than unity would mean more runoff leaves than rain fell. A C of 0.90 at the 100-year event gives 1.125 — use 1.0. Many agencies build this cap into their manual; not all do, and spreadsheets frequently miss it.

Assumptions You Are Accepting

Assumption	Consequence when it fails
Rainfall is uniform over the whole area	Breaks down on large watersheds where a storm cell covers only part of the area
Rainfall duration equals or exceeds T_c	The peak is not reached; the method over-predicts
Peak flow occurs when the whole area contributes	Not true where a small, highly impervious sub-area peaks earlier — check partial-area conditions
C is constant through the storm	Ignores the saturation trend that C_f partially patches
Return period of Q equals that of i	An approximation, not a derivation
No storage anywhere in the system	Ponds, swales and pipe storage all attenuate the peak; the method cannot see them

When You May Use It

Limit	Typical threshold	Note
Drainage area	≤ 200 acres	Many agencies cap far lower — 20 to 50 acres is common; check the local manual, which governs
Output needed	peak flow only	If you need a volume or a hydrograph, use TR-55 / TR-20 / HEC-HMS instead
Storage present	none	Any detention or routing puts you outside the method
T_c	≥ 5 min	Most agencies enforce a 5 or 10 minute floor on the IDF read

The partial-area check. A 30-acre site with a 3-acre parking lot at the outlet can peak higher from the parking lot alone than from the whole site, because the small area's short T_c reads a much higher intensity. Run both and take the larger. This is the failure mode that most often shows up as an undersized inlet.

Sources: Kuichling, E. (1889), the original statement of the method. FHWA HEC-22, Urban Drainage Design Manual, 3rd ed. ASCE/WEF MOP 77, Design and Construction of Urban Stormwater Management Systems. Intensity from NOAA Atlas 14. C_f values as tabulated in HEC-22 and most state DOT drainage manuals — confirm against the manual with jurisdiction over your project, which governs over any general reference including this one.

Card: pe-calc.com/cheat-sheets/rational-method-equation

Fan & Pump Affinity Laws

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 16, 2026

The affinity laws predict how a centrifugal machine — fan, blower or pump — responds to a change in speed, impeller diameter or fluid density. They follow from geometric similarity and constant efficiency, and they are identical in form for fans and pumps. The power law is cubic, which is the entire economic case for variable-speed drives.

Speed Change (same impeller, same fluid)

Quantity	Relation	Ratio form
Flow — Q (cfm or gpm)	Q ∝ N	Q₂/Q₁ = N₂/N₁
Head or pressure — H, Δp	H ∝ N²	H₂/H₁ = (N₂/N₁)²
Shaft power — P	P ∝ N³	P₂/P₁ = (N₂/N₁)³
Efficiency — η	≈ constant	assumed unchanged

What the Cube Law Actually Buys

Speed	Flow	Head / pressure	Power	Power saved
100%	100%	100%	100%	—
90%	90%	81%	72.9%	27%
80%	80%	64%	51.2%	49%
70%	70%	49%	34.3%	66%
60%	60%	36%	21.6%	78%
50%	50%	25%	12.5%	87.5%

Read the 80 percent row: giving up a fifth of the flow cuts power roughly in half. This is why throttling a damper or a valve to trim flow is so wasteful compared with slowing the machine — throttling moves you up the head curve instead of down the power curve.

Impeller Diameter Change (same speed)

Quantity	Relation	Ratio form
Flow	Q ∝ D	Q₂/Q₁ = D₂/D₁
Head	H ∝ D²	H₂/H₁ = (D₂/D₁)²
Power	P ∝ D³	P₂/P₁ = (D₂/D₁)³

The diameter laws are the weaker set. Trimming an impeller does not preserve geometric similarity — the blade exit angle, tip clearance and casing relationship all change. They hold acceptably for trims within roughly 10 to 20 percent of full diameter and drift noticeably beyond that. The speed laws carry no such caveat, which is one more reason to prefer a VFD over a trim where you have the choice.

Density Change (fans and blowers)

Quantity	Relation	Practical effect
Volumetric flow	Q ∝ ρ⁰	Unchanged — a fan moves the same cfm regardless of density
Static pressure	Δp ∝ ρ	Falls with altitude and with hot air
Power	P ∝ ρ	Falls with density — motor sized at sea level is conservative at altitude
Mass flow	ṁ ∝ ρ	The quantity that actually matters for heat transfer and combustion

Fan curves are published at standard air, 0.075 lb/ft³ (roughly 70°F at sea level). At 5,000 ft or in a 400°F flue-gas duct the delivered pressure is materially lower, and selections made straight off the catalogue curve will fall short.

The Trap That Invalidates the Laws

Static head breaks the speed prediction. The affinity laws describe the machine. Where the machine actually lands is the intersection of its curve with the system curve. Only when the system is purely frictional — H ∝ Q², no lift — does the operating point track the affinity relations exactly. Add static lift, as almost every lift station has, and the system curve no longer passes through the origin: flow falls faster than speed, and the cube-law power saving is not realised in full. On a high-static system, slowing a pump too far reaches shut-off head and delivers no flow at all while still drawing power.

Assumptions Behind the Laws

Assumption	Where it fails
Geometric similarity	Trimmed impellers; different casing
Constant efficiency across the change	Large speed turndown moves you off the best-efficiency point
Dynamically similar flow (same Reynolds regime)	Very low speeds; viscous fluids
Incompressible flow	Blowers and compressors above roughly 7 percent pressure rise
No cavitation	NPSH available must still exceed NPSH required at the new point

Sources: Hydraulic Institute Standards (ANSI/HI 14.6) for centrifugal pump affinity relations. AMCA Publication 201, Fans and Systems. ASHRAE Handbook — HVAC Systems and Equipment, fan chapter. Karassik et al., Pump Handbook. The relations are exact consequences of similarity and constant efficiency; the caveats above are where the underlying assumptions, not the algebra, give way.

Card: pe-calc.com/cheat-sheets/fan-pump-affinity-laws

Retaining Wall Design Criteria — Reference

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 24, 2026

The code minimums and design-guide criteria for cantilever, gravity, segmental block, geogrid-reinforced and soil nail walls on one card. Local amendments and owner standards (DOTs, municipalities) often tighten these — check the governing document.

IBC §1807.2.3 — Code minimum for all retaining walls

Check	Static	With earthquake loads
Sliding	1.5	1.1
Overturning	1.5	1.1

§1807.2.1 also requires stability against excessive foundation pressure and water uplift. Common practice for gravity and cantilever walls is 2.0 on overturning, with the resultant within the middle third (e ≤ B/6) on soil.

IBC Table 1610.1 — Minimum lateral soil load (equivalent fluid pressure)

Backfill	USCS	Active (psf/ft)	At-rest (psf/ft)
Well-graded clean gravels; gravel-sand mixes	GW	30	60
Poorly graded clean gravels; gravel-sand mixes	GP	30	60
Silty gravels, poorly graded gravel-sand mixes	GM	40	60
Clayey gravels, poorly graded gravel-clay mixes	GC	45	60
Well-graded clean sands; gravelly sand mixes	SW	30	60
Poorly graded clean sands; sand-gravel mixes	SP	30	60
Silty sands, poorly graded sand-silt mixes	SM	45	60
Sand-silt-clay mix with plastic fines	SM-SC	45	100
Clayey sands, poorly graded sand-clay mixes	SC	60	100
Inorganic silts and clayey silts	ML	45	100
Mixture of inorganic silt and clay	ML-CL	60	100
Inorganic clays of low to medium plasticity	CL	60	100
Organic silts; elastic silts; high-plasticity clays; organic clays	OL, MH, CH, OH	Unsuitable as backfill

Moist soil at optimum density. Saturated or submerged backfill: buoyant soil weight plus full hydrostatic pressure. Walls free to deflect use active; walls restrained at the top (e.g. basement walls braced by a floor) generally use at-rest — see §1610.1 for the exact conditions. A geotechnical investigation may replace these values.

Rankine earth pressure coefficients (level backfill, vertical smooth wall)

φ	K_a	K₀ = 1 − sinφ	K_p	K_aγ at 120 pcf (psf/ft)
26°	0.390	0.562	2.56	47
28°	0.361	0.531	2.77	43
30°	0.333	0.500	3.00	40
32°	0.307	0.470	3.25	37
34°	0.283	0.441	3.54	34
36°	0.260	0.412	3.85	31
38°	0.238	0.384	4.20	29
40°	0.217	0.357	4.60	26

K_a = tan²(45° − φ/2) · K_p = tan²(45° + φ/2) · P_a = ½K_aγH² at H/3 + K_aqH at H/2

Segmental (modular block) walls — NCMA-based minimums

Check	Min. FoS
Base sliding / overturning	1.5 / 2.0
Bearing capacity — soil footing / concrete footing	2.0 / 3.0
Global stability	1.3
Tensile overstress (geogrid)	1.2
Pullout	1.5
Max. unreinforced height (sliding / overturning)	1.5 / 2.0
Facing shear between units / facing connection	1.5 / 1.5
Uncertainties factor	1.5

As adopted in the Town of Clayton, NC Segmental Block Retaining Wall Design standard, Table 2.1 (critical structures), which follows the NCMA Design Manual for Segmental Retaining Walls. The same standard sets a 2,000 psf minimum bearing capacity and a 1% differential settlement limit (evaluate for walls 10 ft and taller).

Geogrid-reinforced walls — geometry and pullout

Item	Value	Source
Minimum reinforcement length (MSE walls)	0.7H	FHWA-NHI-10-024 §4.2
Minimum length, all but highway loading	0.6H	Clayton NC Table 2.1
Minimum length, AASHTO highway loading	0.7H or 8 ft, greater	Clayton NC Table 2.1
Scale-effect factor α, geogrids / geotextiles (no test data)	0.8 / 0.6	FHWA-NHI-10-024 Table 3-6
K_r/K_a for geosynthetic (extensible) reinforcement	1.0, constant with depth	FHWA-NHI-10-024 §4.4

Design surcharges (live load on the retained side)

Use above the wall	Surcharge (psf)
Landscaping walls	0
Pedestrian traffic, light storage	50
Light traffic, auto parking	100
Highway loading, heavy traffic	250
FHWA minimum traffic surcharge (MSE walls)	2 ft of soil

Tiered walls: an upper wall set back less than about 2× the lower wall's height (face to face) surcharges the lower wall and must be included in its design (Clayton NC Table 2.1, note 1).

Soil nail walls — FHWA GEC 7 Table 5.1 (ASD)

Limit state	Static	Seismic
Overall stability (1.35 for some non-critical permanent walls)	1.5	1.1
Overall, temporary excavation lift	1.25–1.33	—
Basal heave (short term / long term)	2.0 / 2.5	2.3
Pullout	2.0	1.5
Lateral sliding	1.5	1.1
Bar tension, Grade 60/75 \| Grade 95/150	1.8 \| 2.0	1.35 \| 1.50
Facing flexure / punching shear	1.5	1.1

Measured maximum nail force (GEC 7 §5.4): 0.5–1.1 K_aγHS_vS_h in the upper two-thirds, averaging about 0.75; the lower third carries about half that. Presumptive bond strengths by soil and drilling method: see the soil nail calculator.

Cantilever wall — first-trial proportions

Dimension (H = total height)	First trial
Base width, B	0.5H – 0.7H
Toe length	≈ B/3
Footing thickness	≈ 0.1H, 12″ min
Stem at base / at top	≈ 0.08–0.1H / 8–12″
ACI 318-19 min. flexural steel (Gr 60) / max. spacing	0.0018A_g / min(3h, 18″)

Sources: International Building Code §1807.2 and §1610, Table 1610.1 (ICC); FHWA-NHI-14-007, Soil Nail Walls Reference Manual (GEC 7, 2015), Table 5.1 and §5.4; FHWA-NHI-10-024, MSE Walls and Reinforced Soil Slopes, Vol. I (2009); Town of Clayton, NC, Segmental Block Retaining Wall Design (2010), Table 2.1, following the NCMA Design Manual for Segmental Retaining Walls; ACI 318-19 Table 7.6.1.1 and §7.7.2.3. Proportions are customary first trials, not requirements.

Card: pe-calc.com/cheat-sheets/retaining-wall-design-criteria

Design speed (mph)	SSD (ft)	K (design)
15	80	10
20	115	17
25	155	26
30	200	37
35	250	49
40	305	64
45	360	79
50	425	96
55	495	115
60	570	136
65	645	157
70	730	181
75	820	206
80	910	231

Design speed (mph)	SSD (ft)	K (design)
15	80	10
20	115	17
25	155	26
30	200	37
35	250	49
40	305	64
45	360	79
50	425	96
55	495	115
60	570	136
65	645	157
70	730	181
75	820	206
80	910	231

All 38 reference cards, one print job

Manning's n — Roughness Coefficient Reference

Lined / Constructed Channels & Pipes

Earth & Excavated Channels

Natural Streams (Bank-full, Top of Bank)

Floodplain (Overbank Flow)

Hazen-Williams C — Pipe Roughness Reference

Plastic Pipe

Iron & Steel Pipe

Concrete & Other

When to use Hazen-Williams vs. Darcy-Weisbach

Rational Method Runoff Coefficient (C) — Reference

By Surface Type

By Land Use

By Lawn / Soil / Slope

Composite (Weighted) C for a Mixed Site

NRCS Curve Number (CN) — TR-55 Reference

Urban / Developed Land Uses (TR-55 Table 2-2a)

Agricultural & Natural Land Uses (TR-55 Table 2-2c)

Hydrologic Soil Groups (HSG)

Composite (Weighted) CN for a Mixed Watershed

Weir Discharge Coefficients — Reference

Sharp-Crested (Suppressed) Rectangular Weir

Broad-Crested Weir / Spillway

V-Notch (Triangular) Weir — Free Discharge

Cipolletti (Trapezoidal) Weir

SI form (metric) of the same equations

Time of Concentration Methods — Comparison

Decision matrix — pick the right method

The equations, side-by-side (US units)

Kirpich (1940)

NRCS Lag (SCS)

TR-55 Segmental (3-segment travel time)

Kerby / Hathaway (overland)

FAA

Common mistakes

Minor Loss Coefficients (K) — Reference

Entrances & Exits

Bends & Elbows

Tees

Valves (Fully Open)

Sudden Expansion & Contraction

Pipe Absolute Roughness (ε) — Darcy-Weisbach Reference

Absolute Roughness by Material

Friction Factor Equations (Turbulent, Re > 4000)

Culvert Entrance Loss Coefficients (Ke) — HDS-5

Concrete Pipe

Corrugated Metal Pipe (CMP)

Box Culvert (Reinforced Concrete)

NRCS / SCS 24-Hour Storm Distributions

The Four Distributions

Geographic Applicability

How It Is Used

Open Channel Geometry — Reference

Section Property Formulas

Manning's Equation

Best Hydraulic Section (Max Q for Given Area)

Orifice Discharge Coefficients (Cd) — Reference

Discharge Coefficient by Orifice Type

Component Coefficients (Sharp-Edged)

Hydraulic Jump — Reference

Core Equations (Rectangular Channel)

Jump Classification by Upstream Froude Number

Riprap Sizing Methods — Reference

Velocity-Based D50 (Isbash)

D50 Riprap Size Chart by Velocity (Isbash, Ss = 2.65)

Gradation Limits (Well-Graded Riprap)

Layer Thickness & Filter

Specific Energy & Critical Depth — Reference

Core Equations

Flow State by Depth

Alternate Depths

Storm Sewer Design Criteria — Reference

Velocity & Slope

Manning's n (Storm Sewer Pipe)

Size, Cover & Storm

Stokes Settling Velocity — Reference

Settling Velocity by Regime

Typical Settling Velocities (quartz, 20°C water)

Reynolds Number & Flow Regimes — Reference

Culvert Entrance Loss Coefficients (K_e) — HDS-5

Orifice Discharge Coefficients (C_d) — Reference

D50 Riprap Size Chart by Velocity (Isbash, S_s = 2.65)

Bottom Bars, Grade 60 — l_d in inches (rounded up)

Failure-Mode Factor K_o and Side Slopes

Von Thun & Gillette Offset C_b

K₁ — pier nose shape

K₂ — angle of attack

K₃ — bed condition

Design speed (mph)	SSD (ft)	K (design)
15	80	10
20	115	17
25	155	26
30	200	37
35	250	49
40	305	64
45	360	79
50	425	96
55	495	115
60	570	136
65	645	157
70	730	181
75	820	206
80	910	231