Register to enable "Calculate" button. Demonstration mode for 0.9 m < D < 1.1 m and N<3 where D is pipe diameter, N is number of pipes (cookies must be enabled).
Diagram of Flow through a Culvert
Links on this page: Introduction
Equations Variables
Values of Coefficients and Manning n Error
Messages and Validity References
In the calculation above:
• Culvert Types:
"Conc. Sq edge. Wall" = Concrete pipe with square edged inlet
and headwall
"Conc. Groove. Wall" = Concrete pipe with groove end at inlet
and headwall
"Conc. Groove. Proj" = Concrete pipe with groove end
projecting at inlet
"CMP. Headwall" = Corrugated metal pipe with headwall at
inlet
"CMP. Mitered" = Corrugated metal pipe mitered to slope at
inlet
"CMP. Projecting" = Corrugated metal pipe projecting at inlet
• Units:
m=meters, ft=feet, l/s=liter/sec, cfm=cubic feet per minute, cfs=cubic feet per second,
gpm=US gallons per minute, gph=US gallons per hour, gpd=US gallons per day, MGD=Millions
of US gallons per day.
• You can enter tailwater depth (Y_{t}) as a negative
number if flow from the culvert drops down to a receiving channel. You don't need to
know the exact elevation drop; entering any negative number for Y_{t}
will have the same effect.
• The phrase "Inlet Control" or "Outlet Control" that
appears in the upper right hand corner of the calculation refers to the type of control
for the total flow (Q_{t}) entered in the calculation's upper left hand
corner. The graph below the calculation plots headwater depth (Y_{h})
for the range of minimum Q_{t} to maximum Q_{t} entered under the word "GRAPHING:" in the
calculation. The type of control may change from one part of the graph
to another as Q_{t} changes.
Introduction
Culverts have been utilized for thousands of years as a means to transmit water under
walkways or roads. Often, a culvert is simply installed without much thought to how
much water it needs to convey under extreme conditions. If a culvert cannot convey
all of the incoming water, then the water will flow over or around the pipe, or simply
back up behind the culvert creating a pond or reservoir. If any of these conditions
are unacceptable, then the proper culvert diameter and number of culverts must be selected
prior to installation in order to convey all of the anticipated water through the pipe(s).
This calculation helps the designer size culverts as well as present a headwater
depth vs. discharge rating curve.
The LMNO Engineering calculation is primarily based on the methodology presented in
Hydraulic Design of Highway Culverts by Normann (1985) and published by the U.S.
Department of Transportation's Federal Highway Administration. It is also known as
HDS5 (Hydraulic Design Series No. 5). HDS5 focuses on culvert design.
Culvert design is usually based on the maximum acceptable discharge  thus the HDS5
methodology is geared toward culverts flowing full with water possibly flowing over the
road above the culvert. In addition to programming the HDS5 methodology, LMNO
Engineering wished to compute headwater depths for lesser flows. Therefore, in
addition to the HDS5 methodology, we have added the Manning equation for culverts flowing
partially full. The HDS5 methodology also assumes that the user knows the tailwater
depth (Y_{t}) before using the methodology. Though Y_{t}
can be found by field measurements, it is often computed in the office using Manning's
equation based on bottom width, side slopes, channel roughness, and channel slope.
Therefore, LMNO Engineering added the additional feature of a builtin subroutine for
computing Y_{t} for trapezoidal channels. Note that for the
graphing portion of our calculation, Y_{t} is recomputed for the entire
range of flows (Q_{t}) shown on the graph (unless the user specifically
inputs Y_{t}).
As explained in Normann, 1985 (also known as HDS5, Hydraulic Design of Highway Culverts), the discharge through a culvert is
controlled by either inlet or outlet conditions. Inlet control means that flow
through the culvert is limited by culvert entrance characteristics. Outlet control means
that flow through the culvert is limited by friction between the flowing water and the
culvert barrel. The term "outlet control" is a bit of a misnomer because
friction along the entire length of the culvert is as important as the actual outlet
condition (the tailwater depth). Inlet control most often occurs for short, smooth,
or greatly downward sloping culverts. Outlet control governs for long, rough, or slightly
sloping culverts. The type of control also depends on the flowrate. For a given culvert
installation, inlet control may govern for a certain range of flows while outlet control
may govern for other flowrates. If the flowrate is large enough, water could go over
the road (or dam). In this case, the calculation automatically computes the amount
of water going over the road and through each culvert, as well as the headwater depth.
If you have surfed around our website, you may have noticed our other calculations for
circular culverts. We have a calculation using Manning's
equation for design of circular culverts. Since it uses Manning's equation, it
assumes the culvert is long enough so that normal depth is achieved. We also have a
calculation for computing discharge from the
exit depth ("end depth") in a circular culvert  very useful for flow rate
measurement in the field. For flows under pressure, we have several calculations
listed under the Pipe Flow category on our home page.
Equations and Methodology
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The LMNO Engineering methodology generally follows that of Normann (1985; also known as
HDS5  Hydraulic Design of Highway Culverts). However, the Normann methodology is mainly for culvert design.
Culvert design usually involves the largest expected flowrate. We wanted to write a
calculation that also determines headwater depth for small flowrates. Therefore, in
addition to the Normann methodology, we have incorporated Manning's equation for outlet
control when the headwater depth is less than 0.93 times the culvert diameter. 0.93D
is used since it is the depth at which discharge through a partially full circular culvert is a
maximum (Chow, 1959). At depths greater than 0.93D and for full flow, the Nomann
(1985) equation is used for outlet control. For inlet control, our calculation uses
Normann's equations.
Many of the equations shown below are empirical and require U.S. Customary units (feet,
seconds, and radians). Some of the equations are based on first principles and are
compatible with any consistent set of units (e.g. SI). However, to keep this web
page from being "too busy", we have refrained from indicating which equations
are empirical and which are fundamental. If you work through the equations by hand,
please use feet, seconds, and radians in all of them, to avoid any problem with units.
Our calculation (above) allows many different types of units; the units are
internally converted before and after using the equations.
Since total flow rate (Q_{t}) is entered and headwater depth (Y_{h})
is computed, the equations below are solved simultaneously to determine Y_{h}.
Outlet versus inlet control is determined by the equation resulting in a larger value for Y_{h}.
All of the variables are defined below in the Variables
section. Pipe downstream invert elevation is defined as 0.0.
General Equations
Q_{t} = Q_{r} + N Q_{p} S_{p} = S_{c}
 Y_{f} /L_{p} E_{i} = L_{p} S_{p}
E_{h} = E_{i} + Y_{h}
V=Q_{p} /A_{v}
Tailwater Depth, Y_{t}
Y_{t} can be computed or input. If it is computed, Manning's
equation is used (Chow, 1959):
Since Q_{t} is input, the above equations are solved numerically
(backwards) for Y_{t}.
Headwater depth, Y_{h}
Y_{h} is computed independently based on inlet and outlet control
equations. The equation that gives the larger value of Y_{h} is
considered to be the controlling mechanism and is reported.
Inlet Control (see below
for values of constants C_{1}, C_{2}, C_{3}, C_{4} , C_{5})
Outlet velocity (V) is computed based on what we call the velocity depth, Y_{v}.
Normann (1985) suggests computing Y_{v} using the Manning equation.
If Y_{v} is greater than D, then Y_{v} is set
to D.
Unsubmerged Inlet (Normann, 1985):
Submerged Inlet (Normann, 1985):
Outlet Control
Outlet velocity (V) is computed based on what we call the velocity depth, Y_{v}.
Normann (1985) suggests: If Y_{t}≤Y_{c}, then Y_{v}=Y_{c}.
If Y_{c}<Y_{t}<D, then Y_{v}=Y_{t}.
If Y_{t}≥D, then Y_{v}=D.
If Y_{h}<0.93D, then
Manning's equation (Chow, 1959) is used:
Since Q_{p} is input, the above equations are solved numerically for Y_{h}.
If Y_{h}≥0.93D, Normann
(1985) is used:
Flow over Road (or Dam)
If water flows over the road (or dam), then flow over the road is computed by (Normann,
1985):
Note that, instead of using a constant value of 3, Normann (1985) uses a
coefficient that varies from 2.5 to 3.1 depending on the water depth above the road and
whether the road is paved or gravel.
Variables
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A=Flow area [ft^{2}].
A_{c}=Flow area in one pipe based on critical depth [ft^{2}].
A_{v}=Flow area in one pipe used for computing outlet velocity [ft^{2}].
b=Width of channel bottom [ft]. Used for computing Y_{t}.
C_{1}, C_{2}, C_{3}, C_{4}, C_{5}=Constants
for inlet control equations. See values below.
D=Diameter of each pipe (culvert) [ft].
E_{h}=Headwater elevation relative to invert of pipe outlet [ft].
Pipe outlet invert elevation is defined at 0.0 ft.
E_{i}=Elevation of pipe inlet invert relative to pipe outlet invert [ft].
Pipe outlet invert elevation is defined at 0.0 ft.
E_{r}=Elevation of road (or dam) crest relative to pipe outlet invert
[ft]. Pipe outlet invert elevation is defined at 0.0 ft.
g=Acceleration due to gravity, 32.174 ft/s^{2}.
H=Head loss computed from outlet control equation [ft].
HW=Abbreviation for headwater.
K_{e}=Minor loss coefficient for pipe inlet (used for outlet control
equations). See values below.
L_{p}=Pipe (culvert) length [ft]. If there is more than one
culvert, they all must have the same length. L_{p} is the length of
one of them (not the sum of the lengths).
L_{w}="Weir" length [ft]. Length of the road (or dam)
that water could flow over. L_{w} is the width that the water sees
as it flows over the road.
n_{c}=Channel Manning n coefficient. See values
below.
n_{p}=Pipe (culvert) Manning n coefficient. See
values below.
N=Number of pipes (culverts) next to each other.
P=Wetted perimeter [ft].
Q_{p}=Flowrate (discharge) through each pipe [cfs, ft^{3}/s].
Q_{r}=Flow rate (discharge) over the road (or dam) [cfs].
Q_{t}=Total flowrate (discharge) [cfs]. Sum of flows through pipes plus flow
over road.
S_{c}=Slope of existing channel [elevation change/length].
Longitudinal slope, not side slopes.
S_{p}=Pipe slope [elevation change/length]. Longitudinal slope, not
side slopes.
T_{c}=Top width of flow in one pipe based on critical depth [ft].
V=Pipe outlet velocity [ft/s].
V_{c}=Pipe velocity based on critical depth [ft/s].
Y_{avg}=Average water depth [ft].
Y_{c}=Critical water depth [ft].
Y_{f}=Fall [ft]. Vertical distance that inlet pipe invert is
lowered below the existing channel bottom.
Y_{h}=Headwater depth [ft].
Y_{o}=Water outlet depth [ft].
Y_{t}=Tailwater depth [ft]. Depth of water in existing channel at
culvert outlet.
Y_{v}=Depth used for computing outlet velocity [ft].
z_{1}=Left side slope of existing natural channel [horizontal/vertical].
z_{2}=Right side slope of existing natural channel [horizontal/vertical].
π=Greek letter "pi" = 3.1415926...
θ=Greek letter "theta". Used for partially full pipes.
Values of Coefficients and Manning n
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Manning n values are from Chow (1950), French (1985), Mays (1999), Normann (1985), and
Streeter (1998). C_{1} through C_{5} and K_{e} are from
Normann (1985).
Pipe material and inlet type 
_{Manning n} 
C_{1} 
C_{2} 
C_{3} 
C_{4} 
C_{5} 
K_{e} 
Concrete. Square edge inlet with headwall. 
0.013 
0.0098 
2.0 
0.5 
0.0398 
0.67 
0.5 
Concrete. Groove end inlet with headwall. 
0.013 
0.0078 
2.0 
0.5 
0.0292 
0.74 
0.2 
Concrete. Groove end projecting at inlet. 
0.013 
0.0045 
2.0 
0.5 
0.0317 
0.69 
0.2 
Corrugated metal (CMP). Headwall at inlet. 
0.022 
0.0078 
2.0 
0.5 
0.0379 
0.69 
0.5 
Corrugated metal (CMP). Mitered to slope at inlet. 
0.022 
0.0210 
1.33 
0.7 
0.0463 
0.75 
0.7 
Corrugated metal (CMP). Projecting at inlet. 
0.022 
0.0340 
1.50 
0.5 
0.0553 
0.54 
0.9 
Channel Material 
Manning n 
Material 
Manning n 
Natural Streams 
Excavated Earth Channels 
Clean and Straight 
0.030 
Clean 
0.022 
Major Rivers 
0.035 
Gravelly 
0.025 
Sluggish with Deep Pools 
0.040 
Weedy 
0.030 


Stony, Cobbles 
0.035 
Floodplains 
Pasture, Farmland 
0.035 
Heavy Brush 
0.075 
Light Brush 
0.050 
Trees 
0.15 
Error Messages and Validity
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calculation
Input checks on left side of calculation. If one of these
messages appears, the calculation and graphing is halted.
"Need 0≤Q_{t}<10000 m^{3}/s". Total flow
cannot be negative or must be less than 10,000 m^{3}/s.
"Need 0<N<1001". Must have at least one pipe, but no more
than 1000 pipes.
"Need 0<D<100 m". Pipe diameter must be positive and less
than 100 m.
"Need 0<Lp<10000 m". Pipe length must be positive and less
than 10,000 m.
"Need 0<Pipe n<0.05". Pipe Manning n must be positive and
less than 0.05.
"Need Y_{t}<E_{r}". Tailwater depth cannot be
higher than the road crest.
"Need E_{i}+D<E_{r}". Upstream pipe invert
plus culvert diameter cannot exceed road crest elevation. If E_{i}+D
is greater than E_{r}, then the top of the culvert is pushing through the
road, which is unacceptable.
"Need 0<L_{w}<10000 m". "Weir" length of
road (or dam) must be positive and less than 10,000 m.
"Need Y_{t}<10000 m". Tailwater depth must be less
than 10,000 m. Negative values are acceptable. Negatives simulate culverts
discharging to a lower channel.
"Need S_{c}<0.5". Channel bottom slope cannot exceed
0.5 m/m (vertical to horizontal ratio). This is the longitudinal slope, not the side
slopes.
"Need S_{c}>0". Channel cannot be horizontal.
"Need 0<Chan n<0.5". Channel Manning n must be positive and
less than 0.5.
"Need 0<b<10000 m". Channel bottom width must be positive
and less than 10,000 m.
"Need 0<z_{1}<10000", "Need 0<z_{2}<10000".
Channel side slopes can be neither exactly vertical (z=0) nor nearly flat
(z>10,000). z is defined as horizontal to vertical ratio.
"Need 1e7<S_{p}<0.5". Pipe slope must be between
these limits.
Input checks for graph and chart (table). If one of these messages
appears, the graph and table will not proceed. Note that if any value is out of range in the
upper portion of the calculation, a graph will not be shown.
"Need min Q_{t}≥0". Minimum total flow for graph was
entered as a negative number.
"Max Q_{t}>10000 m^{3}/s". Maximum total flow
for graph cannot exceed 10,000 m^{3}/s.
"Min must be < Max". Minimum Q_{t} entered
for graph must be less than maximum Q_{t} entered for graph.
"Need Min/Max<0.99". Minimum Q_{t} entered
for graph must be less than 0.99 times maximum Q_{t} entered for graph.
Otherwise, the minimum and maximum are too close together to have good axis labels for the
graph.
Runtime errors. The following message may be generated
by the graphing/table portion of the calculation:
"Y_{t}>E_{r} for some Q_{t}".
Tailwater depth exceeds road (or dam) crest for large values of Q_{t}.
Y_{h} cannot be computed or graphed when Y_{t}>E_{r}
since the equations are only valid for Y_{t}≤E_{r}.
References
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Chow, V. T. 1959. OpenChannel Hydraulics. McGrawHill, Inc. (the classic text)
French, R. H. 1985. OpenChannel Hydraulics. McGrawHill Book Co.
Mays, L. W. editor. 1999. Hydraulic design handbook. McGrawHill Book
Co.
Normann, J. M. 1985. Hydraulic design of highway culverts. HDS5
(Hydraulic Design Series 5). FHWAIP8515. NTIS publication PB86196961.
Obtainable at https://www.fhwa.dot.gov/engineering/hydraulics/library_arc.cfm?pub_number=7&id=13.
Streeter, V. L., E. B. Wylie, and K. W. Bedford. 1998. Fluid Mechanics.
WCB/McGrawHill. 9ed.
© 20012017 LMNO Engineering, Research, and
Software, Ltd. All rights reserved.
Please contact us for consulting or other questions.
LMNO Engineering, Research, and Software, Ltd.
7860 Angel Ridge Rd. Athens, Ohio 45701 USA Phone: (740) 7072614
LMNO@LMNOeng.com http://www.LMNOeng.com

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Critical Depth in Circular Culvert
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