GEOLOGICAL SURVEY CIRCULAR 376

COMPUTATION OF PEAK DISCHARGE 

AT CULVERTS
COMPUTATION OF PEAK DISCHARGE AT CULVERTS

By RW. Carter

GEOLOGICAL SURVEY CIRCULAR 876

Washington, D. C., 1967

Free on application to the Geological Survey, Washington 25, D. C.
United States Department of the Interior

STEWART 1. UDALL, SECRETARY

Geological Survey

THOMAS B. NOLAN, DIRECTOR

REMUNT1D 

IMS

Free on app/konbn to fhe U.S. Geological Survey, Washington 25, 0. C.
CONTENTS

Page

Abstract.............................. 1

Introduction........................... 1

Recent investigations of culvert hydraulics. 1

Classification of principal flow types..... 1

General classification................ 2

Computation of types I, II, and III flow . 2

Type I flow. ....................... 3

Circular sections ................ 4

Rectangular sections ............. 7

Type II flow ....................... 7

Type HI flow ...................... Id

Computation of type IV flow ........... 11

Classification of types V and VI flow ... 11

Concrete culverts.................. 11

Corrugated-pipe culverts ........... 11

Computation of type V flow.......... 14

Computation of type VI flow ......... 14

Unusual conditions ..................... 14

Coefficients of discharge ............... 14

Types I, II, and III flow .............. 16

Flush setting in vertical headwall.... 16

Pipe culverts. ................... 16

Box culverts .................... 17

Wingwall entrance ................. 18

Pipe culverts set flush with vertical headwall. ................... 18

Box culverts .................... 18

Projecting entrance ................ 18

Corrugated-metalpipes and arches 18

Concrete pipes with beveled end ... 18 

Mitered pipe set flush with sloping

embankment...................... 18

Page

Types IV and VI flow .................. 19

Flush setting in vertical headwall. .... 19

Box or pipe culverts............... 19

Wingwall entrance .................. 19

Pipe culverts set flush with vertical 

headwall........................ 19

Box culverts ..................... 19

Projecting entrance ................ 19

Corrugated-metal pipes and arches . 19 

Concrete pipes with beveled end .... 19

Mitered pipe set flush with sloping 

embankment....................... 20

Type V Flow. ......................... 20

Flush setting in vertical headwall..... 20

Pipe or box culverts .............. 20

Wingwall entrance .................. 20

Pipe culverts set flush with vertical 

headwall.......................'. 20

Box culverts ..................... 20

Projecting entrance ................. 20

Corrugated metal pipes and arches . 20 

Concrete pipes with beveled end .... 20

Mitered pipe set flush with sloping 

embankment....................... 21

Roughness coefficients .................. 21

Collection of field data .................. 21

Culvert geometry..................... 21

Head wate r. .......................... 21

Tail water ..................... v ... . 21

Application to field problems. ............ 22

Bibliography ........................... 26

Symbols ................... Inside back cover

ILLUSTRATIONS

Plate 

Figure

9.

10.

11.

12.

13.

14.

15.

16.

Page 

Classification of culvert flow ..................................... Inside back cover

Procedure for classification and computation of culvert flow...................... 3

Relation between specific energy and depth ..................................... 3

Critical slope as a function of head for culverts of circular section, with free outfall 4 

Relation between head and critical depth in culverts of circular section............ 7

Critical slope for culverts of rectangular section, with free outfall................ 7

Relation between height of head-water surface and height of water surface at culvert

inlet, culverts of circular section ........................................... 9

Relation between height of head-water surface and height of water surface at culvert

inlet, culverts of rectangular section. ........................................ 10

Criteria for types V and VI flow in box or pipe culverts with concrete barrels and

square, rounded, or beveled flush entrances from a vertical headwall, either with

or without wingwalls........................................................ 12

Criteria for types V and VI flow in pipe culverts with rough barrels ............... 13

Relation between head and discharge for type VI flow ............................ 15

Base coefficient of discharge for types I, II, and III flow in pipe culverts with

square entrance mounted flush with vertical headwall.......................... 16

Variation of the discharge coefficient with entrance rounding, types I, II, and III

flow in box or pipe culverts set flush with vertical headwall .................... 17

Variation of the discharge coefficient with entrance beveling, types I, II, and III

flow in box or pipe culverts set flush with vertical headwall .................... 17

Base coefficient of discharge for types I, II, and III flow in box culverts with square

entrance mounted flush in vertical headwall .................................. 18

Variation of discharge coefficient with wingwall angle, types I, II, and HI flow in

box culverts with wingwall set flush with sloping embankment.................. 18

Variation of discharge coefficient with ratio of head-water height to pipe diameter,

types I, H, and IH flow in mitered pipe set flush with sloping embankment ....... 19

iii
TABLES

Page 

Table 1. Characteristics of flow types.................................................. 2

2. Coefficients of area, hydraulic radius, conveyance, and discharge for criticaldepth flow in pipes of circular section with part-full flow. ...................... 5

3. Properties of circular pipes .................................................. 8

4. Discharge coefficients for box or pipe culverts set flush in a vertical headwall,

types IV and VI flow........................................................ 19

5. Discharge coefficients for box or pipe culverts set flush in a vertical headwall,

type V flow. ............................................................... 20

6. Discharge coefficient for box culverts with wingwalls, type V flow ................ 20
COMPUTATION OF PEAK DISCHARGE AT CULVERTS

By R.W. Carter

ABSTRACT

- Methods for computing peak flood flow through 

culverts on the basis of a field survey of highwater marks and culvert geometry are presented. 

These methods are derived from investigations 

of culvert flow as reported in the literature and 

on extensive laboratory, studies of culvert flow. 

For convenience in computation, culvert flow 

has been classified into six types, according to 

the location of the control section aud the relative heights of the head-water and tail-water 

levels. The type of flow which occurred at any 

site can be determined from the field oat a and 

the criteria given in this report. A discharge 

equation has been developed for each flow type 

by combining the energy and continuity equations 

for the distance between an approach section upstream from the culvert and a terminal section 

within the culvert barrel. The discharge coefficient applicable to each flow type is listed for 

the more common entrance geometries. Procedures for computing peak discharge through 

culverts are outlined in detail for each of the 

six flow types.

INTRODUCTION

The culvert is frequently used as an indirect 

means of determining peak discharge from 

small drainage areas. Computation of discharge 

based on high-water marks is used in lieu of a 

direct measurement by the current-meter 

method because the short time interval between 

rainfall and runoff does not permit the engineer 

to reach the site with the equipment necessary 

for a direct measurement. Also, the short duration of the flood and poor measuring conditions 

would often result in a current-meter measurement of questionable accuracy.

The purpose of this report is to set forth 

procedures for computation of peak discharge 

through culverts. This report supersedes the 

manual Computation of Peak Discharge 

Through Culverts by S. E. Rantz, M. A. 

Benson, and J. S. Cragwall, Jr., which is 

presently used in the Survey. The methods 

outlined represent no radical departure from 

those presently used. Rather, they feature 

simplified methods of computation by use of 

formulas instead of flow routing, better definition of the discharge coefficient by laboratory investigation, and extension of the methods 

to include culverts that are under a high he ad.

Geological Survey Circular 284 (Kindsvater, 

Carter, and Tracy, 1953) defined procedures for 

computing peak discharge through bridge openings. It differs from the present report primarily in the range of variables involved: a 

culvert may generally be distinguished from a 

bridge waterway by a length-to-width ratio 

greater than one.

This manual was prepared as part of the research program of the Surface Water Branch, 

Water Resources Division, U. S. Geological 

Survey. The author was aided in the laboratory 

investigation by Harry H. Barnes and Jacob 

Davidian. C. E. Kindsvater, Professor of Civil 

Engineering, Georgia Institute of Technology, 

served as a consultant to the research program.

RECENT INVESTIGATIONS OF 

CULVERT HYDRAULICS

In preparing this manual full use has been 

made of previously published investigations on 

the discharge characteristics of culverts. Since 

1920, the U. S. Bureau of Public Roads has 

sponsored many laboratory investigations at 

colleges and universities throughout the country. Notable among them is the report by 

Yarnell, Nagler, and Woodward (1926) which 

contains the result of more than 3, 000 tests on 

flow through pipe and box culverts. These tests 

included a range of pipe diameters from 12 to 

30 inches and a range of box sizes from 2 to 4 

feet. Various entrance geometries were tested, 

but only flow with the culvert barrel submerged 

on both ends was investigated.

F. T. Mavis (1942) reported on a laboratory 

investigation on round smooth pipes. His tests 

included a comprehensive range of head-water 

and tail-water elevations, but only an entrance 

form with a sharp edge set flush in a headwall 

was tested.

Straub and Morris (1950a-c) reported on fullscale tests on corrugated and concrete-pipe 

culverts. This work was extended in a later report by Straub, Anderson, and Bowers (1954).

Other investigations to develop culvert geometries of greater hydraulic efficiency are listed 

in the bibliography. These reports have also 

been used in defining coefficients for certain 

entrance geometries.

In addition to the reports cited, the procedures 

given in this manual are based on a laboratory 

investigation conducted by the U. S. Geological 

Survey at the Georgia Institute of Technology. 

The tests included all conditions of flow through 

round, rectangular, and square barrels with 

various entrance geometries. Sand-grain 

roughness was used to obtain a range in the 

friction factor (Darcy I) of from 0. 017 to 0. 085 

for barrel diameters ranging from 4 to 6 inches. 

The tests were run in a flume 2 feet wide, the 

bed slope (of which was varied from 0 to 8 percent.

CLASSIFICATION OF PRINCIPAL FLOW TYPES

In defining the discharge characteristics of 

culverts, the energy equation is written for a
COMPUTATION OF PEAK DISCHARGE AT CULVERTS

reach that begins at an approach section upstream 

from the culvert and ends at a terminal section 

within the culvert barrel. Whether the terminal 

section is at the upstream or downstream end of 

the culvert depends upon the type of flow in the 

barrel. If critical or rapid flow occurs at the 

culvert entrance, the height of the head-water 

surface is not a function of either the barrel 

friction loss or .the height of the tail-water surface, and the terminal section is thus located at 

the upstream end of the culvert. For ail other 

types of flow the terminal section is located at 

the downstream end of the culvert.

Information obtained in th-e field survey includes the peak elevation of the water surface

upstream and downstream from the culvert and 

the geometry of the culvert and approach channel. Reliable high-water marks can rarely be 

found in the culvert barrel; therefore, the type 

of flow that occurred during the peak flow cannot always be determined directly from field 

data, and classification becomes a trial-anderror procedure.

General Classification

For convenience in computation, culvert ftow 

has been classified into six types on the basis 

of the location of the control section and the 

relative heights of the head-water and tail-water 

surfaces/ Table 1 lists the characteristics-of

TABLE 1, --Characteristics of flow types

[D = maximum vertical heiglrt of barrel and diameter of circular 

culverts; see also plate 1]

Flow 

type

I. ...........

II. ...........

III. ...........

IV.. ...........

V. ...........

VI. ...........

Barrel 

flow

Partly full

- - do

- - do - -

Full

Partly full

Full

Location of 

terminal 

section

Inlet

Outlet

- - do

- - do

Inlet

Outlet

Kind of 

control "

depth 

- - do - -

- - do - -

geometry

and 

barrel 

geometry

Culveri 

slope !

Mild

- - do - -

An IT

hv - z

< 1.5 |

<1.5

<1.5

>1.0

>1.5

fl.5

*4 '.

fe- :

< l.,0

<i.«

>1,0

*£

D

< 1.0

< 1.0

^ 1.0

> 1.0

< 1.0

^ 1.0

the six types, and plate 1 illustrates the types of 

flow. From these the following general classification can be made.

1. If h4/D is equal to or less than 1.0 and 

(h l z)/D is less than 1.5, only types I, II, and 

III flow are possible.

2. If h^/D is greater than 1.0, only type IV 

flow is possible.

3. If h4/D is equal to or less than 1. 0 and 

(h l z)/D is equal to or greater than 1.5, only 

types V and VI flow are possible.

The type of flow may thus be narrowed to 1 of 

3 groups, as shown by figure 1. Outlined on this 

figure are the steps necessary for identifying the 

type of flow that occurred. The following sections of this report are organized in the same 

manner as the illustration so that the procedure 

may be followed in computing the discharge for 

any culvert.

One exception to the general classification 

must be noted. Type I flow can occur with h4 

slightly greater than he or with h+/D greater

than 1. 0 if the bed slope is very steep. When these 

conditions occur, negative results will be obtained from the formulas given for the general 

classification, and methods outlined under Unusual Conditions should be used.

Computation of Types I, II, and III Flow

Flow types I, II, and III are identified by the 

ratio of the bed slope of the culvert to the critical 

slope and the ratio of the critical depth to the 

tail-water depth. As shown by plate 1, type III 

flow will occur if the elevation of the downstream 

water surface is higher than the water-surface 

elevation at critical depth. If the tail-water elevation is lower than the water-surface elevation 

at critical depth, type I flow will occur if the 

bed slope of the culvert is greater than the 

critical slope, or type II flow will occur if the 

bed slope is less than the critical slope.

To determine which type of flow occurred, 

(1) assume that type I flow occurred, (2) compute 

the elevation of the water surface at critical 

depth and the critical slope, and (3) compare the 

critical slope with the bed slope and the watersurface elevation at critical depth with the
CLASSIFICATION OF PRINCIPAL FLOW TYPES

GENERAL CLASSIFICATION

TYPE I,H,OR HI

ASSUME TYPE I

TYPE H OR HI

TYPE 31 OR

FINAL 

CLASSIFICATION

Figure 1. --Procedure for classification and computation of culvert flow.

tail-water elevation. This will result in positive 

identification of types I or III flow or narrow the 

possible flow types to types II or III.

Type I Flow

If type I flow is assumed, the critical depth 

should be at the inlet. Critical depth, d , is the 

depth at the point of minimum specific energy 

for a given discharge and cross section. The 

relation between specific energy and.depth is 

illustrated in figure 2. The specific energy, ft , 

is the height of the energy grade line above the 

lowest point in the cross section. Thus,

H d ~t~ " "

5 rf

where

Ho = specific energy, 

d = maximum depth in the section, 

V = mean velocity in the section, and 

g - acceleration of gravity.

It can be shown that at the point of minimum 

specific energy and critical depth, rfc ,

<?' A3

Figure Z. --Relation between specific energy 

and depth.

and

Yl- d -A. ~ -

where

Q = discharge in cfs,
COMPUTATION OF PEAK DISCHARGE AT CULVERTS

A = areaof section below the water surface,

T = width of the section at the water surface,

dc - maximum depth of water in the criticalflow section, and

dm = mean depth in section = A/T.

For the condition of minimum specific energy 

and critical depth, the discharge equation for a 

section of any shape can be written

(la)

(Ib)

The discharge equation can be simplified according to the shape of the sections.

Thus, for rectangular sections,

<? = S.67 b dc3/* 

and, for circular sections,

0 = c q

(2)

(3)

where

b - width of section,

cfl = function of dg/D as given in table 2 

(column 5), and

dc = maximum depth of water in the criticalflow section.

If critical depth occurs at the inlet, the discharge may be computed with the applicable 

critical -depth equations 1, 2, or 3, and the energy equation 4 as written for the distance from 

an approach section to the inlet,

(4)

where

C = the discharge coefficient,

K, = the mean velocity in the approach section,

h f = the head loss due to friction between 

1-2 the approach section and the inlet =

L, <X?VK,KC ;,

K = conveyance = (1.486 JP2/3 A)/n , and

Lw = distance from the approach section to 

the inlet.

Normally a critical depth is assumed which 

fixes the value of the remaining unknown terms. 

A good first approximation is dc = 0.66(hl z). 

Successive approximations of dc will quickly 

converge toward the solution.

In order to check the assumption of type I 

flow, the critical slope for the culvert is computed as S = (Q/Kc)*. Here, Sc is the critical 

slope and KC is the conveyance of the section of 

critical depth at the inlet. If S C<S 0 and b4<h 

the assumption of type I flow has been proved 

and the correct discharge has been computed. 

If /i>Ai , type III flow has been identified and 

the discharge may be computed as outlined on 

page 10. If h4<hc and Sc>S 0 , type II flow is 

assumed and the analysis is continued as outlined on page 7.

For circular and rectangular sections, the 

identification of flow type is simplified by figures 3, 4, and 5. The procedure is outlined 

below.

1.4 

1.3

1.2

h,-z ' '

D

0.9

0.8

0.7

0.5

0.4 1

TYPE

i

/

f

n -

/

^^~s«

/

s

 Se

/

/

^^

4

^

PEI

^

'"'

 ~- ->.

Dotum/

r- ' ir-^^

7~

SO 40 50 60 | 70

«"L

80 *

Figure 3. --Critical slope as a function of head 

for culverts of circular section, with free outfall.

Circular sections

1. Compute (h l - z)/D and (So Dl/3)/n2 

and plot the point on figure 3.

A point to the right of the curve indicates type I 

flow and a point to the left type II flow.

2. Determine dc from figure 4.

3. Compare (dc + z) with h4 .
CLASSIFICATION OF PRINCIPAL FLOW TYPES

TABLE 2. --Coefficients of area (Ac), 'hydraulic radius (Rc), conveyance (Kc )» and discharge 

(qc) for critical-depth flow in pipes of circular section and with part-full flow

[d = maximum depth of water, in feet; D = diameter of pipe, in feet]

i

D

0.01

.02

.03

.04

.05

.06

.07

.08

.09

.10

ill

.12

.13

. 14

.15

.16

.17

.18

.19

.20

.21

.22

.23

.24

.25

.26

.27

.28

.29

.30

.31

.32

.33

.34

.35

.36

.37

.38

.39

.40

.41

.42

.43

.44

.45

.46

.47

.48

.49

.50

ca

A e = C.D'

0.0013

.0037

.0069

.0105

.0147

.0192

.0242

.0294

.0350

.0409

.0470

.0534

.0600

.0668

.0739

.0811

.0885

.0961

.1039

. 1118

.1199

.1281

.1365

.1449

.1535

.1623

.1711

.1800

.1890

.1982

.2074

.2167

.2260

.2355

.2450

.2546

.2642

.2739

.2836

.2934

.3032

.3130

.3229

.3328

.3428

.3527

.3627

.3727

.3827

.3927

' c.

*c - C,D

0. 0066

.0132

.0197

.0262

.0325

.0389

.0451

.0513

.0575

.0635

.0695

.0755

.0813

.0871

.0929

.0985

.1042

.1097

.1152

.1206

.1259

.1312

.1364

.1416

.1466

.1516

.1566

.1614

.1662

.1709

.1756

.1802

.1847

.1891

.1935

.1978

.2020

.2062

.2102

.2142

.2182

.2220

.2258

.2295

.2331

.2366

.2401

.2435

.2468

.2500

C*

K _ c °8/3

0. 000068

. 000307

. 000747

.001376

. 002228

. 00328

. 00457

.00601

. 00775

. 00966

.0118

.0142

.0168

.0195

.0225

.0257

.0291

.0327

.0366

.0405

.0446

.0491

.0537

.0586

.0634

.0685

.0740

.0792

.0848

.0907

.0968

.1027

.1088

.1155

.1220

.1283

.1350

.1421

.1488

.1561

.1631

.1702

.1780

.1854

.1931

.2002

.2080

.2160

.2235

.2317

C«

Q.c = C D5/a

0. 0006

.0025

.0055

.0098

. 0153

.0220

.0298

.0389

.0491

.0605

.0731

.0868

.1016

.1176

.1347

.1530

.1724

.1928

.2144

.2371

.2609

.2857

.3116

.3386

.3666

.3957

.4259

.4571

.4893

.523

.557

.592

.628

.666

.704

.743

.784

.825

.867

.910

.955

1.000

1.046

1.093

1.141

1.190

1.240

1.291

1.343

1.396
COMPUTATION OF PEAK DISCHARGE AT CULVERTS

TABLE 2.--Coefficients of area (Ac) hydraulic radius (Rc ) conveyance (Kc ) and discharge for 

(qc ) critical-depth flow in pipes of circular section and with part-full flow--Continued

[d = maximum depth of water, in feet; D - diameter of pipe, in feet]

dc 

D

0.51

.52

.53

.54

.55

.56

.57

.58

.59

.60

.61

.62

.63

.64

.65

.66

.67

.68

.69

.70

.71

.72

.73

.74

.75

.76

.77

.78

.79

.80

.81

.82

.83

.84

.85

.86

.87

.88

.89

.90

.91

.92

.93

.94

.95

.96

.97

.98

.99

1.00

c a

A c = CaD>

0.4027

.4127

.4227

.4327

.4426

.4526

.4625

.4724

.4822

.4920

.5018

.5115

.5212

.5308

.5404

.5499

.5594

.5687

.5780

.5872

.5964

.6054

.6143

.6231

.6319

.6405

.6489

.6573

.6655

.6736

.6815

.6893

.6969

.7043

.7115

.7186

.7254

.7320

.7384

.7445

.7504

.7560

.7612

.7662

.7707

.7749

.7785

.7817

.7841

.7854

cr

* c a CD

0.2531

.2562

.2592

.2621

.2649

.2676

.2703

.2728

.2753

.2776

.2799

.2821

.2842

.2862

.2882

.2900

.2917

.2933

.2948

.2962

.2975

.2987

.2998

.3008

.3017

.3024

.3031

.3036

.3039

.3042

.3043

.3043

.3041

.3038

.3033

.3026

.3018

.3007

.2995

.2980

.2963

.2944

.2921

.2895

.2865

.2829

.2787

.2735

.2666

.2500

C*

'.-'^ c * n

0.2394

.2472

.2556

.2630

.2710

.2791

.2873

.2955

.3031

.3115

.3192

.3268

.3346

.3423

.3501

.3579

.3658

.3727

.3805

.3874

.3953

.4021

.4090

.4157

.4226

.4283

.4349

.4415

.4470

.4524

.4578

.4630

.4681

.4731

.4768

.4816

.4851

.4884

.4916

.4935

.4951

.4966

.4977

.4979

.4970

.4963

.4940

.4902

.4824

.4633

C,

0.c = C0S/>

1.449

1.504

1.560

1.616

1.674

1.733

1.792

1.853

1.915

1.977

2.041

2. 106

2.172

2.239

2.307

2.376

2.446

2.518

2.591

2.666

2.741

2.819

2.898

2.978

3.061

3. 145

3.231

3.320

3.411

3.505

3.602

3.702

3.806

3.914

4.028

4. 147

4.272

4.406

4.549

4.70

4.87

5.06

5.27

5.52

5.81

6. 18

6.67

7.41

8.83

--
CLASSIFICATION OF PRINCIPAL FLOW TYPES

1.6

1.4

12

h,- z

7

D

LO

0.8

0.6

0.4

0.2 0.3 0.5 0.6 0.7 OB

Figure 4. --Relation between head and critical depth in culverts of circular section.

4. Type I flow can occur only if the criterion 

of step 1 is met and h4 < (dc + z). If type I 

flow is identified, compute the discharge with equations 1 or 3 and 4 as outlined on page 4.

Rectangular sections

1. Compute the parameters of figure 5 for the 

culvert, assuming that dc = 0. 66 f/i, - z), and 

plot the point. A point to the right of the line indicates type I flow, atid a point to the left indicates type II flow.

2. Compare (dc + z) with h4 .

3. Type I flow can occur only if the criterion 

of step 1 is met and h4 < (dc + z). If type I flow is 

identified, compute the discharge with equations 

1 or 2 and 4 as outlined above.

Type II Flow

If type II flow is assumed, the critical depth 

should occur at the outlet. The flow equations

 5 .03 

.02 

.01 

0

^/

/

T1

/

PE n

/

> + -

/

^

/

. T

/

TPE I

/^

/

Crilia

/

1 slop.

/

t.Sc

3 .01 .02 .03 .04 .05 .O 

So

Figure 5. --Critical slope for culverts of rectangular section, with free outfall.
COMPUTATION OF PEAK DISCHARGE AT CULVERTS 

TABLE 3. --Properties of circular pipes

Diameter 

(inches)

6

8

10

12

15

18

21

24

30

36

42

48

54

60

66

72

78

84

96

108

120

132

144

156

168

180

Diameter, Z> 

(feet)

0.500

.667

.833

1.00

1.25

1.50

1.75

2.00

2.50

3.00

3.50

4.00

4.50

5.00

5.50

6.00

6.50

7.00

8.00

9.00

10

11

12

13

14

15

Area, A O 

(square feet)

0.196

.349

.545

.785

1.23

1.77

2.41

3.14

4.91

7.07

9.62

12.6

15.9

19.6

23.8

28.3

33.2

38.5

50.3

63.6

78.5

95.0

113

133

154

177

Z>2

0.250

.444

.694

1.00

1.56

2.25

3.06

4.00

6.25

9.00

12.2

16.0

20.2

25.0

30.2

36.0

42.2

49.0

64.0

81.0

100

121

144

169

196

225

0S/2

0. 177

.363

.634

1.00

1.75

2.76

4.05

5.66

9.88

15.6

22.9

32.0

43.0

55.9

71.0

88.2

108

130

181

243

316

401

499

609

733

871

D9/3

0. 157

.339

.614

1.00

1.81

2.95

4.44

6.34

11.5

18.7

28.1

40.3

55.1

73.0

94. 1

119

147

179

256

351'

464

599

755

935

1140

1370

used for the computation of type I flow are also 

applicable here, with the further provision that 

the barrel-friction loss must be accounted for in 

the energy equation since the control section has 

shifted to the outlet.

Thus,

(5)

The discharge and the critical depth must be 

computed by solution of equation 5 and the applicable critical-depth equations 1 and 2 or 3. 

The solution is tedious, because in order to

compute the barrel-friction loss, h

2.3 the

height of the water surface at the inlet must be 

established. Therefore, the procedure for computation of discharge outlined below is recommended.

1. Assume the critical depth at the outlet; a 

good first assumption is df = 0.66 ft..

2. Compute the corresponding discharge with 

the applicable critical-depth equations 1 and 

2 or 3.

3. Compute the depth of water, c/2 

inlet with the energy equation.

at the

26 C2

hi - z (6)

for the distance from the approach section to 

the inlet, neglecting the approach velocity head 

and the friction loss between the approach section and the inlet. Here the depth, d2 , is 

measured from the lowest point in the section. 

The equation may be solved by successive approximation. Graphical solutions of equation 

6 for circular and rectangular sections are presented in figures 6 and 7, respectively. The 

depth at the inlet may be computed directly from 

these graphs.

4. Compute the conveyance of the section at 

the approach, the inlet, and the outlet.
CLASSIFICATION OF PRINCIPAL FLOW TYPES

Figure 6. Relation between height of head-water surface and height of water surface at culvert inlet

for culverts of circular section.
10 COMPUTATION OF PEAK DISCHARGE AT CULVERTS

.65

.70

.75

.80

£L

Elevation

.85

.90

SQUARE AND .RECTANGULAR 

SECTIONS

.95

1.00 .02 .04 .06 .08 .12 .14 .16

2g(h,-

Figure 7. --Relation between height of head-water surface and height of water surface at culvert inlet

for culverts of rectangular section.

5. Compute the friction loss between the ap 

proach and the inlet,

and between the inlet and the outlet,

6. Compute the velocity head, V^/lg.

7. Compute the discharge with equation 5.

8. If the discharge computed with equation 5 

is not equal to the discharge computed in step 2, 

the assumed value of d is incorrect. The correct value of d must fie determined by successive approximation, repeating the procedure 

outlined above.

After the discharge and the elevation of the 

water surface at critical depth are established, 

the assumption of type II flow is checked by 

comparing the elevation of critical depth with 

the tail -water elevation. If h > h4 , type II flow 

occurred. If Jic < h. , type III flow occurred and 

the discharge may *"» computed as outlined in 

the next section.

Type III Flow

In type III flow, critical depth does not occur in 

the culvert, and the upstream elevation of the water 

surface for a given discharge is a function of the 

surface elevation of the tail water. Watersurface elevations h l and 7i4 can normally be 

established from high-water marks, and it is 

assumed that /i 3 = /i4 . The discharge can be 

computed from the energy equation

0 =

The following procedure is recommended in 

computing discharge:

1. Assume a discharge.

2. Determine the depth at the inlet, d2 . by 

trial solution of equation 6, or directly from 

figures 6 or 7 if the culvert is circular or rectangular in section.

3. Compute the conveyance of the sections at 

the approach, the inlet, and the outlet.
CLASSIFICATION OF PRINCIPAL FLOW TYPES 11

4. Compute the friction loss between the approach section and the inlet,

and between the inlet and the outlet,

J. = / (-¥-> 

*'... L K,K3 '

5. Compute the velocity head, V^/lg.

6. Compute the discharge with equation 7.

7. If the discharge computed with equation 7 

is not equal to the assumed discharge, then 

another discharge should be assumed and the 

procedure outlined above repeated.

Computation of Type IV Flow

In type IV flow, both the entrance and the outlet are submerged. The discharge may be computed directly from the energy equation for the 

distance between section 1 and section 4. In deriving the discharge formula shown below, the 

velocity head at section 1 and the friction loss 

between sections 1 and 2 have been neglected, 

and the energy loss between sections 3 and 4 is 

assumed to beV2 3/2g.The velocity head at section 4 is also assumed to be zero. Thus,

h. = h. + 1 4

h + h

V2

O = CA /2g(h, - hj 

Y 1+ 29C*n^

(8)

Classification of Types V and VI Flow

Type V or VI flow will occur if the tail water 

is below the crown at the outlet, and (h l z)/D 

is equal to or greater than 1.5.

As shown in plate 1, part-full flow under a 

high head is classified as type V. The flow 

pattern is similar to that downstream from a 

sluice gate with supercritical flow near the 

entrance. The occurrence of type V flow requires a relatively square entrance that will 

cause contraction of the area of live flow to less 

than the area of the culvert barrel. In addition, 

the combination of barrel length, roughness, 

and bed slope must be such that the contracted 

jet will not expand to the full area of the barrel. 

If the water surface of the expanding flow comes 

in contact with the top of the culvert, type VI 

flow will occur because the passage of air to the 

culvert will be sealed off, causing the culvert to 

flow full throughout its length; under these conditions, the head-water surface drops, indicating a more efficient use of the culvert barrel.

Within a certain range either type V or type VI 

flow may occur, depending upon factors that are 

very difficult to evaluate. For example, the 

wave pattern superimposed on the water-surface 

profile through the culvert can be important in 

determining full or part-full flow. For these and 

other reasons the criteria for identifying type V 

or type VI flow must be considered approximate.

Concrete Culverts

Figure 8 may be used to classify type V or VI 

flow in concrete culvert barrels by the procedure 

outlined below.

1. Compute the parameters L/D, r/D, or 

w/D, and So

2. Select the curve of figure 8 corresponding 

to f/D or w/D for the culvert. Sketch in an 

interpolated curve for the given r/D or w/D, if 

necessary.

3. Plot the point defined by S o and L/D for 

the culvert.

4. If the point plots to the right of the curve 

selected in step 2, the flow was type VI; if the 

point plots to the left of the curve, the flow was 

type V.

The use of figure 8 is restricted to square, 

rounded, or beveled flush entrances from a 

vertical headwall, either with or without wingwalls. Wingwalls do not affect the flow classification, as the rounding effect they provide is 

offset by a tendency to produce vortices that 

supply air to the culvert entrance. For culverts 

with wingwalls use the geometry of only the top 

side of the entrance in computing the effective 

radius of rounding, r, or the effective bevel, 

w, in using figure 8.

Corrugated-Pipe Culverts

Figure 9 may be used to classify type V or VI 

flow in rough pipes mounted flush with a vertical 

headwall, either with or without wingwalls, as 

outlined below.

1. Determine the ratio r/D for the pipe.

2. From figure 9 select the graph corresponding to the value of r/D for the culvert.

3. Compute the ratio 29n 2 (h^ - z)/Ro4/3

and select the corresponding curve on the graph 

selected in step"2. Sketch in an interpolated 

curve for the_computed ratio, if necessary.

4. Plot the point defined by S o and J./D for 

the culvert.

5. If the point plots to the right of the curve 

selected in step 3, the flow was type VI; if the 

point plots to the left of the curve, the flow was 

type V.
12 COMPUTATION OF PEAK DISCHARGE AT CULVERTS

0.09

Figure 8. --Criteria for types V and VI ilow in box or pipe culverts with concrete barrels and square, 

rounded, or beveled flush entrances from a vertical headwall, either with or. without wing walls.
0.08 0.08

0.08 O.OSr

(c)

Figure 9. Criteria for types V and VI flow in pipe culverts with rough barr
14 COMPUTATION OF PEAK DISCHARGE AT CULVERTS

For a pipe projecting from a he ad wall or embankment, figure 9a should be used in classifying the flow.

Computation of Type V Flow

In type V flow the culvert entrance is submerged, and the tail water is below the crown 

at the outlet. The flow is supercritical near the 

entrance to the culvert. The discharge may be 

computed directly from the equation

1 - z)

Computation of Type VI Flow

(9)

In type VI flow both the inlet and the outlet are 

submerged. The discharge equation for the distance between sections 1 and 3, neglecting

and h

1 . 2

(10)

A straightforward application of dischargeequation 10 is hampered by the necessity of deterra^tting /i 3 , which varies from a point below 

th%pe»isfeer of the outlet to its top, even though 

the water surface is at the top of the culvert. 

This variation in piezometric head is a function 

of the Froude number. This difficulty has been 

circumvented by basing the data analysis upon 

dimensionless ratios,

where f\/D is proportional to the Froude number.

The functional relation expressed above has 

been defined by laboratory experiment and is 

presented in figure 10. The term "/&" has been 

deleted from the discharge ratio for convenience. 

The following procedure may be used to compute discharge.

1. Compute the ratio h^/D. Select the discharge coefficient, C , applicable to the culvert 

geometry.

2. From figure 10 determine the value of 

0/4 o vTT corresponding to 2<*n 27,/J? o 4 /3 = 1.

3. Compute the ratio 29n 2/,//? 0 4/3 for the 

culvert under study.

4. From figure 10, using the computed ratio

29n 2 /,/ff 0 4/3 and the coefficient C, find the 

correction factor k f.

5. Multiply the value of Q/A i/D~ from step 2 

by the value of k f from step 4, thus determining an adjusted ratio Q/A o /F . Determine 

the value of Q. from the adjusted ratio.

UNUSUAL CONDITIONS

One exception to the general flow classification, 

already noted, is that type I flow can occur with 

h4 slightly greater than hc or with /i4 greater 

than D. This condition will be recognized when 

negative results are obtained from the formulas 

given for the general flow classification. The 

discharge for this flow condition may be solved 

by trial computation of the backwater curve that 

extends from the known tail-water surface to the 

head-water surface. A unique solution is obtained 

when the computed head-water surface is the 

same as that observed from high-water marks, 

or,

V*

+ 2 + h f + h +

2g r 2.3 1.2

There will be, of course, occasional culvertflow problems of unusual complexity, examples 

of which have not been covered in detail here. In 

most of these, results may be computed by going 

back to the fundamentals of the subject or to 

literature covering the special conditions. Undoubtedly there will be occasional problems incapable of reliable solution.

Examples of unusual conditions that might be 

experienced are culverts of nonuniform barrel 

geometry, culverts with changes in bottom slope 

within the barrel, submerged culverts (type IV 

flow) with flared outlets, culverts with drop 

inlets, rapid flow (mean depth less than critical 

depth) in the approach section, and culverts on an 

adverse slope.

COEFFICIENTS OF DISCHARGE

Coefficients of discharge, C, for'flow types I 

to VI have been defined by laboratory study. The 

coefficients vary from 0.40 to 0.98 and have been 

found to be a function of the degree of channel 

contraction and the geometry of the culvert entrance. In dimensionless notation,

C = f(m, r/D, vr/b, 8, etc.)

where m is a measure of the degree of channel 

contraction, and the other terms within the 

parentheses describe the entrance geometry.

For culverts, the percentage of channel contraction, to, is defined as (1 A/A^) 100 where 

A is the area of flow at the terminal section and 

A I is the area of the approach section. Because 

the value of m is usually large for flood flows, 

the laboratory tests placed emphasis on this condition. However, tests on flow through bridge 

openings as reported by Kindsvater, Carter, and 

Tracy (1953) demonstrate that the discharge coefficient varies almost linearly between values of 

m of 0 and 80, and that the coefficient reaches a 

minimum value at m = 80. All coefficients given 

here are for an m of 80. If the contraction ratio 

is smaller than 80, the value of C may be computed by interpolating between the value of C 

listed for an m of 80 and a value of C of 0. 98 for an m 

of 0. The following formula may be used in lieu of 

interpolation: C(adjusted) = 0. 98 - (0.98 -C ) m/80 .
COEFFICIENTS OF DISCHARGE 15

K) CM 

..=g-OUViJ 398VHDSIO

_J

in

0

N H

!' 

1

0

u

*

-d 

nt

01

00

£

.OI1V8 39HVHOSIC]
16 COMPUTATION OF PEAK DISCHARGE AT CULVERTS

For certain entrance geometries the discharge 

coefficient is obtained by multiplying a base coefficient by an adjustment factor such as k f or 

kw. If this procedure results in a discharge coefficient greater than 0.98, a coefficient of 0.98 

should be used in computing the discharge 

through the culvert.

The coefficients are applicable to both singleand multiple-culvert installations. If the distance between barrels is less than 0. 1 the width 

of a single barrel, its effect should not be considered in determining the effect of the entrance 

geometry.

.Laboratory tests also indicate that the discharge coefficient does not vary with the proximity of the culvert floor to the ground level at 

the entrance. Thus in types I, II, and III flow, 

the geometry of the sides determines the value 

of C ; similarly, in types IV, V, and VI flow the 

value of C varies with the geometry of the top 

and sides. If the degree of rounding or beveling 

is not the same on both sides, or on the sides 

and the.top, the effect of r or w must be obtained by averaging the coefficients determined 

for each of the sides. One exception is noted: if 

the vertical sides of the culvert are rounded or 

beveled and the top entrance is square, multiply 

the average coefficient determined by the procedure just described by 0. 90 for type V flow, 

and by 0. 95 for types IV and VI flow, using the 

coefficient for the square entrance as the lower 

limiting value.

In listing the discharge coefficients, it is convenient to divide the 6 flow types into 3 groups,

each group having a discharge equation of the 

same general form. Thus, flow types I, II, and 

III form one group, types IV and VI another, and 

type V a third. The coefficient C is descriptive 

of the live-stream contraction at the inlet and 

its subsequent expansion in the barrel of the 

culvert. Hence, coefficients for types I, II, and 

III flow should be identical for identical geometries, as should coefficients for types IV and 

VI.

In a systematic presentation of the coefficients, the entrance geometries have been 

classified in four general categories, with necessary subdivision. The 4 classes are common 

to all 3 flow-type groups.

In general, coefficients for flow types I, II, 

III and V are based on tests conducted by the 

Survey at Georgia Institute of Technology, and 

coefficients for flow types IV and VI are based 

primarily on tests by Yarnell and Straub.

 Types I, II, and III Flow 

Flush Setting in Vertical Headwall 

Pipe culverts

The discharge coefficient for square-ended 

pipes set flush in a vertical headwall is a function of 

the ratio of the head-water height to the pipe diameter, (/>! z)/D. The coefficient for flow types I 

II, and III can be determined from figure 11.

If the entrance to the pipe is rounded or beveled, the discharge coefficient is computed by

.90 

.80

70

>

M » 

|^

Xh

Pipe

«. ^

^ + ^

culverts

**, ^

with

^ . ^

^

square

 « >«

 . l Ml

«

^ |^

«l

> ^

^

entrance

 .

^

 

«,

J

«l

>

*

4 .6 .8 1.0 1.2 1.4 l.<

RATIO OF HEADWATER HEIGHT TO PIPE DIAMETER ili 

D

Figure 11. --Base coefficient of discharge for types I, II, and III flow in pipe culverts with square entrance mounted flush with Vertical headwall.
COEFFICIENTS OF DISCHARGE 17

120

LOS

LOC

.02 .04 .06 .06 .10 .12 .14

Figure 12. --Variation of the discharge coefficient with entrance rounding, types I, II, and III flow in

box or pipe culverts set flush with vertical headwall.

multiplying the coefficient for the square-ended 

pipe by an adjustment factor, kf or kw. These 

adjustment factors are a function of the degree 

of entrance rounding or beveling, and these relations as applicable to flow types I, II, and III 

are defined in figures 12 and 13.

The radius of rounding of corrugated metal 

pipes and pipe arches is about three-eighths inch. 

The bevel, w, on all types of concrete pipe, including the bellmouthed pipe, is measured as 

(Dm D)/2 where Dm is the maximum inside diameter at the entrance (see figure 13) and D is 

the diameter of the barrel.

Box culverts

The discharge coefficient for box culverts set 

flush in a vertical headwall is a function of the 

Froude number. The Froude number for flow 

types I and II is always 1.0 and the corresponding discharge coefficient is 0.95. The discharge 

coefficient for type III flow can be determined 

from figure 14 after computing the Froude number, V/Sgd, at the downstream end of the culvert.

If the entrance to the box is rounded or beveled, the discharge coefficient is computed by 

multiplying the coefficient for the square-ended 

box by an adjustment factor, kf or kw . These 

adjustment factors as applicable to flow types 

I, II, and III can be determined from figure 12 

or 13, respectively.

1.20 

I.IS

K.

I.IO

.05

^<f////////X '*^//////7/Y//l

.02 .04 .06 .08 .10

JL D

Figure 13. --Variation of the discharge coefficient with entrance beveling, types I, II, and 

III flow in box or pipe culverts set flush with 

vertical headwall.
18 COMPUTATION OF PEAK DISCHARGE AT CULVERTS

1.00

«0

« 0

Box culvert with square entrance

.5 .6 

FROUDE NUMBER, F =

.8 .9 1.0

Figure 14. --Base coefficient of discharge for types I, II, and III flow in box culverts with square entrance mounted flush in vertical headwall.

Wingwall Entrance 

Pipe culverts set flush with vertical headwall

The addition of wingwalls to the entrance of 

pipes set flush in a vertical headwall does not 

affect the discharge coefficient, which can be 

determined from pages 16 and 17.

Box culverts

The discharge coefficient for box culverts with 

a wingwall entrance is computed by first selecting a coefficient from figure 14 and then multiplying this coefficient by an adjustment factor, 

K8 , which can be determined from figure 15 on 

the basis of the angle 6 of the wingwall. For the 

special condition of 8 = 90°, use KB = 1.06. If 

the angle of the wingwall is not the same on each 

side, the value of C for each side should be determined independently, and the results averaged.

Projecting Entrance 

Corrugated-metal pipes and arches

The discharge coefficient for pipes and pipe 

arches that extend beyond a headwall or embankment may be determined by first computing a 

coefficient as outlined for pipes set flush in a 

vertical headwall and then multiplying this coefficient by an adjustment factor, &L . The adjustment factor is a function of £ _/£ where Lp is 

the length by which the culvert projects beyond 

the headwall or embankment.

L/D

0

.1 

1.0

1.15

Figure 15. --Variation of discharge Coefficient 

with wingwall angle, types I, II, and III flow in 

box culverts with wingwall set flush with sloping embankment.

Concrete pipes with beveled end

The discharge coefficient for projecting entrances is the same as for flush entrances for 

concrete pipes with a beveled end (p. 16).

Mitered Pipe Set Flush With 

Sloping Embankment

The discharge coefficient for mitered pipes 

set flush with a sloping embankment is a function
COEFFICIENTS OF DISCHARGE 19

O O

o tr 

< i 

o 

<n 

o

u. 

o

K 

Z 

UJ 

O

u_ 

u.

UJ 

O

o

CO O

00 O

.70

Mitered pipe

.6 .8 1.0 1.2 1.4 1.6

RATIO OF HEADWATER HEIGHT TO PIPE DIAMETER h'" z

D

Figure 16. --Variation of discharge coefficient with ratio of head-water height to pipe diameter, types 

I, II, and III flow in mitered pipe set flush with sloping embankment.

of the ratio of head-water height to pipe diameter and can be determined from figure 16.

Types IV and VI Flow 

Flush Setting in Vertical Headwall 

Box or pipe culverts

The discharge coefficient for box or pipe culverts set flush in a vertical headwall may be 

selected from table 4. This includes squareended pipes or boxes, corrugated pipes, corrugated-pipe arches, corrugated pipes with a 

standard conical entrance, concrete pipes with 

a beveled or bellmouthed end, and box culverts 

with rounded or beveled sides.

Table 4.--Discharge coefficients for box or pipe 

culverts set flush in a vertical headwall; types 

IV and VI flow

r/b, w/b, w/D, or r/D

O4 ______ _ _ _

08

.12-- ------ -._ ------

C

O O A

QQ

.91

.94

.96 

.97

QQ . Vo

Wingwall Entrance > 

Pipe culverts set flush with vertical headwall

The addition of wingwalls to the entrance of 

pipes set flush with a vertical headwall does not 

affect the discharge coefficient, which can be 

determined from table 4.

Box culverts

For box culverts with wingwalls and a square 

top entrance the discharge coefficient = 0.87. If 

the top entrance is rounded or beveled, select a 

coefficient from table 4 on the basis of the value 

of w/D or r/D for the top entrance but using 

0.87 as a lower limiting value.

Projecting Entrance 

Corrugated-metal pipes and arches

The discharge coefficient for corrugatedmetal pipes and arches that extend past a headwall or embankment may be determined by first 

selecting the coefficient from table 4 that corresponds to the particular value of r/D and then 

multiplying this coefficient by an adjustment 

factor kL (p. 18).

Concrete pipes with beveled end

The discharge coefficient for concrete pipes 

with a beveled end that have a projecting
20 COMPUTATION OF PEAK DISCHARGE AT CULVERTS

entrance is the same as for those with a flush 

entrance and can be determined from table 4.

Mite red Pipe Set Flush With 

Sloping Embankment

The discharge coefficient for pipes mitered 

and set flush with a sloping embankment is 

0.74.

Type V Flow

Flush Setting in Vertical Headwall 

Pipe or box culverts

The discharge coefficient for box or pipe culverts 

set flush in a vertical headwall can be determined 

from table 5. This includes the square-ended pipe 

orbox, corrugated pipe, the corrugated-pipe arch,

TABLE 5.--Discharge coefficients for box or pipe culverts set flush in a vertical headwall; type V flow

Value of C when r/b, w/b, r/D, or w/D equals-

(/»! - Z)/D

1.4 ..............

1.5 ..............

1.6 ..............

1.7 ..............

1.8 ..............

1.9 ..............

2.0 ..............

2.5 ..............

3.0 ..............

3.5 ..............

4.0 ..............

5.0 ..............

0

0.44

.46

.47

.48

.49

.50

.51

.54

.55

.57

.58

.59

0.02

0.46

.49

.51

.52

.54

.55

.56

.59

.61

.62

.63

.64

0.04

0.49

.52

.54

.55

.57

.58

.59

.62

.64

.65

.66

.67

0.06

0.50

.53

.55

.57

.58

.59

.60

.64

.66

.67

.68

.69

0.08

0.50

.53

.55

.57

.58

.60

.61

.64

.67

.69

.70

.71

0. 10

0.51

.54

.56

.57

.58

.60

.61

.65

.69

.70

.71

.72

0.14

0.51

.54

.56

.57

.58

.60

.62

.66

.70

.71

.72

.73

concrete pipe with a beveled end, and the box 

culvert with rounded or beveled sides.

Wingwall Entrance 

Pipe culverts set flush with vertical headwall

For pipes set flush with a vertical headwall, 

the addition of wingwalls to the entrance does 

not affect the discharge coefficient, which can 

be determined from table 5.

Box culverts

The discharge coefficient for box culverts with 

wingwalls and a square top entrance can be determined from table 6. If the top entrance is rounded 

or beveled, the coefficient should be selected from 

table 5 on the basis of w/D or r/D for the top entrance, 

but using the coefficient from table 6 as a lower 

limiting value.

TABLE 6. --Discharge coefficients for box culverts with wingwalls; type V flow

Value of C when the angle of 8 of wingwall is----

(/ij - z)/a

1.3 ..........................

1.4 ..........................

1.5 ..........................

1.6 ..........................

1.7 ..........................

1.8 ..........................

1.9 ..........................

2.0 ..........................

2.5 ..........................

3.0 ..........................

3.5 ..........................

4.0 ..........................

5.0 ..........................

30°

0.44

.46

.47

.49

.50

.51

.52

.53

.56

.58

.60

.61

.62

450

0.44

.46

.47

.49

.50

.51

.52

.53

.56

.58

.60

.61

.62

60°

0.43

.45

.46

.48

.48

.50

.51

.52

.54

.56

.58

.59

.60

750

0.42

.43

.45

.46

.47

.48

.49

.49

.52

.54

.55

.56

.58

90°

0.39

.41

.42

.43

.44

.45

.46

.46

.49

.50

.52

.53

.54

Projecting Entrance 

Corrugated-metal pipes and arches

The discharge coefficient for pipes and pipe 

arches that extend,past a headwall or embankment is determined by first selecting a coefficient from table 5 and then multiplying by an adjustment factor kL (p. 18).

Concrete pipes with beveled end

The discharge coefficient for concrete pipe 

with a beveled end can be determined from table 5.
COLLECTION OF FIELD DATA 21

Mite red Pipe Set Flush With Sloping 

Embankment

The discharge coefficient for mitered pipes 

set flush with a sloping embankment can be 

determined by first selecting a coefficient from 

table 5 for a square-ended pipe and then multiplying this coefficient by 0. 92.

ROUGHNESS COEFFICIENTS

The following values of Manning's coefficient, 

n, are recommended for use in computing the 

conveyance of culvert sections.

Material n

Concrete:

Very smooth. ..................... 0. 012

Ordinary field construction......... .015

Badly spalled ..................... . 020

Corrugated metal ................... . 024

Cement rubble ............... . 020- . 030

Cast iron .......................... . 013

Vitrified pipe ....................... .013

The composite value of n for corrugatedmetal pipe with paved invert may be computed 

by the formula

= Pp X Q.Q12 + (P - Pp X 0.024; 

P

where 

Pp - length of wetted perimeter that is 

paved, and

= total length of wetted perimeter. 

COLLECTION OF FIELD DATA

A transit survey of floodmarks and culvert 

geometry should be made as soon after the flood 

as possible, and the following information 

should be obtained.

Culvert Geometry

A complete survey is necessary to determine 

the dimensions and elevation of the barrel, 

headwalls, bed slope, .and alinement of the culvert. The material used in the culvert should 

be noted, as well as the condition of the material. Values of n for the culvert barrel and 

the approach reach to the culvert should be recorded. Stereographs of the culvert showing all 

pertinent conditions upstream and downstream 

from the culvert should be obtained.

Head Water

The location and elevation of floodmarks along 

the embankment and upstream from the culvert 

should be obtained.

If the area of the approach channel to the culvert is estimated as equal to or greater than 

five times the area of the culvert barrel, a 

zero approach velocity in the approach section 

may be assumed, and no approach section is 

necessary. Otherwise a cross section of the 

channel should be at a distance upstream 

from the toe of the wingwall equal to the 

width of the entrance to the culvert at the 

toe of the wingwall. The cross section should 

be taken at right angles to the channel. If 

high-water marks cannot be found at the location specified for the approach section, the 

approach section should be taken where highwater marks can be found.

Tail Water

Floodmarks that represent the elevation of 

the water surface downstream from the culvert outlet, and not necessarily the elevation 

of the issuing jet, should be obtained close 

to the outlet. They should preferably be obtained along the embankment close to the 

culvert exit.
22 COMPUTATION OF PEAK DISCHARGE AT CULVERTS 

APPLICATIONS TO FIELD PROBLEMS

EXAMPLE NUMBER 1

CONCRETE BOX CULVERT

Given:

6 . = 45° 

h l * 1.02 ft

z = 1.00ft

 > 

D = 8.00 ft

Computed elsewhere:

Aj = 330 sq ft

Jfj =38.900 

Procedure:

1. Compute (ht-*)/D = 1.00.

2. Compute h^/D = 0.75.

3. From steps 1 and 2 the flow is either type I, II, or III (see table 1).

4. Compute C = 0.95 X 1.13 = 1. 07, assuming type I or II flow.
APPLICATIONS TO FIELD PROBLEMS 23

5. Use C = 0.98.

6. Assume type I flow.

7. Assume dc = 0.66 (h1 - z) = 0.66(8.02) = 5.39.

14.5 n2 (b+ 2. ; 14.SX 0.015 2 (8 + 10.78) 8. Compute ^ *-'- 8(2. 30)13 ^ = 0.0058.

1.00

9. Compute S = = 0.017. 

60

10. From steps 3 and 4 plot point on figure 5. As the point plots to the right of the line, S>SC>

the control is at the inlet, and the flow is type I. Note that h c , 6. 39 ft, is greater than h4 , 

6. 00 ft.

11. Compute Jf = ^r? X 2.182/3 X 43.1 = 7130. / 

K c 0.015

12. Compute Q = 5.67 b <fc3/2 = 5.67 X 8 X 5.393/2 = 569 cfs.

13. Compute ft/i2 = t^l^=34 ^JSL. . ,.04 f«.

V 2 /SfioY8 1

14. Compute -LL- = I-2^! X = 0,05 ft.

2g \330/ 64.3

15. Compute Q = C AcW2gth1 - z +^~~ rfc ~ h f}

(? = 0.98 X 8 X 5.39 ^2^8.02 + .05 - 5.39 - .04) = 552 cfs.

16. As the discharge computed in step 12 is greater than that computed in step 15, the assumed 

value of d is too large.

 17. Assume </c = 5.28 ft.

!£.. Compute 0 = 5.67 X 8 X 5.283/2 = 551 cfs.

19. Compete 1,,^-M (^^ ^ =0.04 ft.

V 2 /551\ 2 1

20. Compute -i- = ( ) X = 0.04 ft.

21. Compute Q = 0.98 X 8 X 5.28 /2^(8.02 + .04 - 5.28 - .04) = 550 cfs.

22. As the discharge computed in step 18 is virtually the same as that computed in step 21, the correct discharge is 550 cfs.

23. As a final check that this is type I flow, repeat steps 8 to 10.
24 COMPUTATION OF PEAK DISCHARGE AT CULVERTS

EXAMPLE NUMBER 2

*£ TSQOl C

o| o

2,r=

34'

"in in

60'

PLAN

f

15.00'

n = 0-015

O6r~ Soo'

ELEVATION

T1

8'

-8M

SECTION

CONCRETE BOX CULVERT

= 1. 00 ft. 

4 = 6. 00 ft. 

= 60. 0 ft. 

= 0.015

Given:

0 =45°

*>! = 15.00ft

* =8. 00 ft. 

w =0 

Procedure:

1. Compute <h i ~ *)/D » 1.75.

2. Compute ^4/^ = 0.75.

3. From table 1 the flow is type V or VI.

4. Compute L/D = 7.5; S = 0.017; w/D = 0.

O '

5. Use the curve of figure 8 marked w/D = Oas the base curve.

6. Plot the point on figure 8 defined by the value of L/D and S . As this point is to the left of 

the curve marked w/D = 0, type V flow occurred. °

7. Select the value of ^ from table 6; 0.50.

8. Compute Q - CA ( 

Q

26 (h l - z) 

0.50 X 8 X 8 y~5g 

Q - 962 cfs.

(14.00)
BIBLIOGRAPHY 25

Karr, M, H., 1953, Model studies of inlet design for pipe culverts on steep grades: Oreg. 

State Coll., Proj. Rept. 145.

Kindsvater, C. E., Carter, R. W., andTracy, 

H. J., 1953, Computation of peak discharge 

at contractions: U. S. Geol. Survey Circ. 

284.

Mavis, F. T., 1942, Hydraulics of culverts: 

Pa. State Coll., Bull. 56.

Straub, L.. G., Anderson, A. G., and Bowers, 

C. E., 1954, Effect of inlet design on capacity of culverts on steep slopes: Minn. 

Univ. Proj. Rept. 37.

Straub, L. G., and Morris, H. M., 1950a, 

Hydraulic tests on concrete pipes: Minn. 

Univ. Tech. Paper 4, Series B.

-----1950b, Hydraulic tests on corrugated 

metal pipe: Minn. Univ. Tech. Paper 5, 

Series B.

______I950c, Hydraulic data comparison of

concrete and corrugated metal culvert pipes: 

Minn. Univ. Tech. Paper 3, Series B.

Yarnell, D. L., Nagler, F. A., and Woodward, 

S. M., 1926, Flow of water through culverts: 

Iowa Univ. Bull. 1.

* U.S. GOVERNMENT PRINTING OFFICE : 1963 O 694-990
SYMBOLS

= area of flow at section designated by 

subs c ript

~ area of culvert barrel; the area between 

barrels is excluded in multiple box 

culverts

= area of section of flow at critical depth

= width of box culvert measured at right 

angles to axis of culvert

= coefficient of discharge

= maximum inside vertical dimension of 

culvert barrel, or the inside diameter of circular section. For corrugated pipes, D is measured as the 

minimum inside diameter of the pipe

= maximum inside diameter, in feet, of 

culvert at entrance

- depth of flow, in feet, as measured to 

the lowest point in the cross section

- critical depth, in feet

= mean depth = A/T

= Darcy friction factor

= Froude number; for rectangular section 

= V/ Jgd

- acceleration of gravity = 32.2 ftpersec 

per sec

= height, in feet, of average piezometric 

head at any section above the invert 

of the culvert at the downstream end. 

Piezometric head or height, in general, is equivalent to pressure head 

plus elevation; for uniform flow in 

open channels, it is equivalent to the 

elevation of the water surface. In 

zones of rapid acceleration, the average piezometric height may not be 

equivalent to the water surface

= head loss, in feet, due to entrance 

contraction

= head loss, in feet, due to friction; ^

is the head loss due to friction between section 1 and section 2 and

equals Lw(Q3/KtK3) 

= specific energy 

- adjustment factor

K = conveyance of a section = (1.486/n)R2/3A

i- = length, infect, of the culvert barrel. For 

miteredpipe, the shortest length is used

£ p = distance, in feet, a culvert barrel pro-, 

jects beyond aheadwall or embankment

L W = distance, in feet, from approach section 

to the entrance of the culvert. The top 

of the pipe is used as the entrance 

point for miter ed pipe

at = percent of the channel contraction; for 

culverts, a is approximated by 

(1 - A/A!) 100, where A is the area 

of flow at the terminal section

n = Manning's roughness coefficient

P = wetted perimeter, infeet, of cross section of flow

Q = discharge, in cubic feet per second 

r - radius, in feet, of entrance rounding

J? = hydraulic radius of section of flow; 

equal to A/P

K0 = hydraulic radius of culvert barrel

S = slope of energy grade line

So = bed slope of culvert barrel, ft per ft

Sc = bed slope of culvert for which the normal depth and the critical depth are 

equal

T = width, in feet, of the section at the 

water surface

w = distance, in feet, from point of beginning of bevel to line of projection of 

culvert barrel; pipe or box (see fig. 

13)

z = S L

o

8 - the acute angle (theta) at which the 

wingwaU and head wall join

< = less than

> = not less than

> = greater than

< = not greater than

Subscripts 1, 2, 3 denote location of section. 

Section 1 is the approach section, section 2 is 

in the culvert barrel .at the entrance, and section 3 is in the culvert barrel at the outlet.
GEOLOGICAL SURVEY CIRCULAR 376 PLATE 1

TYPE

I

CRITICAL DEPTH 

AT INLET

D < ''5

h4 /hc<I.O

So > Sc

II

CRITICAL DEPTH 

AT OUTLET

D

h4 /hc <I.O 

SQ < Sc

III

TRANQUIL FLOW 

THROUGHOUT

h4 /D? 1.0 

h /hc >I.O

EXAMPLE TYPE EXAMPLE

SUBMERGED

OUTLET 

h,-z

h4/D>I.O

RAPID FLOW 

AT INLET

FULL FLOW 

FREE OUTFALL

CLASSIFICATION OF CULVERT FLOW