HY8:Polynomial Coefficients

Overview

For circular, box, elliptical, pipe arch, concrete open-bottom arch (commonly called CON/SPAN), and South Dakota Concrete Box culverts, polynomial coefficients, found in Tables 1-6, are utilized in the inlet control headwater computations. Other culvert shapes use Table 7, which shows the HW/D points A(1) through A(10) for interpolation. Each row of coefficients represents different inlet configurations for different culvert shapes.

Note About Coefficient Changes in HY-8 7.3 and Higher

In HY-8 7.3 and later versions of HY-8, several significant changes were made to the coefficients used in HY-8. A summary of the changes to the HY-8 coefficients in this version follows:

Changes to Shapes Using Polynomial Coefficients

Changed the slope correction coefficient, SR, used for all the mitered inlet configurations to the recommended -0.7

Changes to Box Culverts

Changed the 1.5:1 Bevel Wingwall inlet configuration from HY-8 Equation 6 to equation 2. For HY-8 Equations 2, 3, and 6, added 0.01 to the "A" Coefficient in the shape database to account for the fact that the equations were derived using a 2% slope (a 2% slope was used to derive the polynomial equations, meaning 0.5(0.02) was subtracted from each of the polynomial curves and needed to be added back into the equations before correcting for slopes).

Changes to Shapes using A(1) to A(10) Interpolation Coefficients

Added the slope correction term SR*Slope to the interpolation equations in the code and added 0.01 to the interpolation coefficients for thin, square, and bevel inlets. Subtracted 0.01 for the mitered inlet. Added the SR coefficients (All = 0.5 except for mitered which = -0.7) to the coefficient database and the documentation on this page.

Table 1. Polynomial Coefficients - Circular

HY-8 Equation Inlet Configuration KE SR A BS C DIP EE F

1

Thin Edge Projecting

0.9

0.5

0.187321

0.56771

-0.156544

0.0447052

-0.00343602

8.96610E-05

2

Mitered to Conform to Slope

0.7

-0.7

0.107137

0.757789

-0.361462

0.1233932

-0.01606422

0.00076739

3

Square Edge with Headwall (Steel/Aluminum/Corrugated PE)

0.5

0.5

0.167433

0.538595

-0.149374

0.0391543

-0.00343974

0.000115882

4

Grooved End Projecting

0.2

0.5

0.108786

0.662381

-0.233801

0.0579585

-0.0055789

0.000205052

5

0.2

0.5

0.114099

0.653562

-0.233615

0.0597723

-0.00616338

0.000242832

6

Beveled Edge (1:1)

0.2

0.5

0.063343

0.766512

-0.316097

0.0876701

-0.009836951

0.00041676

7

Beveled Edge (1.5:1)

0.2

0.5

0.08173

0.698353

-0.253683

0.065125

-0.0071975

0.000312451

8

sq. proj.

0.2

0.5

0.167287

0.558766

-0.159813

0.0420069

-0.00369252

0.000125169

9

0.5

0.5

0.087483

0.706578

-0.253295

0.0667001

-0.00661651

0.000250619

10

end sect.

0.4

0.5

0.120659

0.630768

-0.218423

0.0591815

-0.00599169

0.000229287

EQ #'s: REFERENCE

1-9 : Calculator Design Series (CDS) 3 for TI-59, FHWA, 1980, page 60

1-10: Hydraulic Computer Program (HY) 1, FHWA, 1969, page 18

Table 2. Polynomial Coefficients - Embedded Circular

HY-8 Equation Inlet Configuration KE SR A BS C DIP EE F

1

20% Embedded, Projecting End, Pond

1.0

0.5

0.0608834861787302

0.485734308768152

-0.138194248908661

0.027539172439404

-0.00214546773150856

0.0000642768838741702

2

40% Embedded, Projecting End, Pond

1.0

0.5

0.0888877561313819

0.431529135749154

-0.073866511532321

0.0159200223783949

-0.00103390288198853

0.0000262133369282047

3

50% Embedded, Projecting End, Pond

1.0

0.5

0.0472950768985916

0.59879374328307

-0.191731763062064

0.0480749069653899

-0.00424418228907681

0.00014115316932528

4

0.55

0.5

0.0899367985347424

0.363046722229086

-0.0683746513605387

0.0109593856642167

-0.000706535544154146

0.0000189546410047092

5

0.55

0.5

0.074298531535586

0.4273662972292

-0.0849120530113796

0.0157965200237501

-0.00102651687866388

0.0000260155937601425

6

0.55

0.5

0.212469378699735

0.511461899639209

-0.174199884499934

0.0410961018431149

-0.00366309685788592

0.000123085395227651

7

20% Embedded, 45 degree Beveled End

0.35

0.5

0.0795781442396077

0.373319755852658

-0.0821508852481996

0.0148670702428601

-0.00121876746632593

0.0000406896111847521

8

40% Embedded, 45 degree Beveled End

0.35

0.5

0.0845740029462746

0.389113662011417

-0.0685090654986062

0.0117190357464366

-0.000790440416133214

0.0000226453591207209

9

50% Embedded, 45 degree Beveled End

0.35

0.5

0.0732498224366533

0.426296207882289

-0.0825309806843494

0.0158108288973248

-0.00103586921012557

0.0000265873062363919

10

20% Embedded, Mitered End 1.5H:1V

0.9

0.5

0.075018832861494

0.404532870578638

-0.0959305677963978

0.0172402567402576

-0.00121896053512953

0.0000338251697138414

11

40% Embedded, Mitered End 1.5H:1V

0.9

0.5

0.086819906748455

0.362177446931189

-0.048309284166457

0.00870598247307798

-0.000359506993503941

2.89144278309283E-06

12

50% Embedded, Mitered End 1.5H:1V

0.9

0.5

0.0344461003984492

0.574817400258578

-0.204079127155295

0.0492721656480291

-0.00436372397619383

0.000144794982321005

EQ #'s: REFERENCE

1-12: NCHRP 15-24 report

Table 3. Polynomial Coefficients - Box

HY-8 Equation Inlet Configuration KE SR A BS C DIP EE F

1

Square Edge (90 & 15 degree flare) Wingwall

0.5

0.5

0.122117

0.505435

-0.10856

0.0207809

-0.00136757

0.00003456

2

1.5:1 Bevel (19-34 degree flare) Wingwall

0.2

0.5

0.1067588

0.4551575

-0.08128951

0.01215577

-0.00067794

0.0000148

3

0.2

0.5

0.1666086

0.3989353

-0.06403921

0.01120135

-0.0006449

0.000014566

4

Square Edge (30-75 degree flare) Wingwall

0.4

0.5

0.0724927

0.507087

-0.117474

0.0221702

-0.00148958

0.000038

5

Square Edge (0 degree flare) Wingwall

0.7

0.5

0.144133

0.461363

-0.0921507

0.0200028

-0.00136449

0.0000358

6

1:1 Bevel (45 degree flare) Wingwall

0.2

0.5

0.0995633

0.4412465

-0.07434981

0.01273183

-0.0007588

0.00001774

EQ #'s: REFERENCE

1-6: Hydraulic Computer Program (HY) 6, FHWA, 1969, subroutine BEQUA

1,4,5: Hydraulic Computer Program (HY) 3, FHWA, 1969, page 16

1,3,4,6: Calculator Design Series (CDS) 3 for TI-59, FHWA, 1980, page 16

Table 4. Polynomial Coefficients - Ellipse

HY-8 Equation PIPE Inlet Configuration KE SR A BS C DIP EE F

27

CSPE

0.5

0.5

0.01267

0.79435

-0.2944

0.07114

-0.00612

0.00015

28

CSPE

mitered

0.7

-0.7

-0.14029

1.437

-0.92636

0.32502

-0.04865

0.0027

29

CSPE

bevel

0.3

0.5

-0.00321

0.92178

-0.43903

0.12551

-0.01553

0.00073

30

CSPE

thin

0.9

0.5

0.0851

0.70623

-0.18025

0.01963

0.00402

-0.00052

31

RCPE

square

0.5

0.5

0.13432

0.55951

-0.1578

0.03967

-0.0034

0.00011

32

RCPE

grv. hdwl

0.2

0.5

0.15067

0.50311

-0.12068

0.02566

-0.00189

0.00005

33

RCPE

grv. proj

0.2

0.5

-0.03817

0.84684

-0.32139

0.0755

-0.00729

0.00027

EQ #'s: REFERENCE

27-30: Calculator Design Series (CDS) 4 for TI-59, FHWA, 1982, page 20

31-33: Calculator Design Series (CDS) 4 for TI-59, FHWA, 1982, page 22

Table 5. Polynomial Coefficients - Pipe Arch

HY-8 Equation PIPE Inlet Configuration KE SR A BS C DIP EE F

12

CSPA

proj.

0.9

0.5

0.08905

0.71255

-0.27092

0.07925

-0.00798

0.00029

13

CSPA

proj.

0.9

0.5

0.12263

0.4825

-0.00002

-0.04287

0.01454

-0.00117

14

CSPA

proj.

0.9

0.5

0.14168

0.49323

-0.03235

-0.02098

0.00989

-0.00086

15

CSPA

proj.

0.9

0.5

0.09219

0.65732

-0.19423

0.04476

-0.00176

-0.00012

16

CSPA

mitered

0.7

-0.7

0.0833

0.79514

-0.43408

0.16377

-0.02491

0.00141

17

CSPA

mitered

0.7

-0.7

0.1062

0.7037

-0.3531

0.1374

-0.02076

0.00117

18

CSPA

mitered

0.7

-0.7

0.23645

0.37198

-0.0401

0.03058

-0.00576

0.00045

19

CSPA

mitered

0.7

-0.7

0.10212

0.72503

-0.34558

0.12454

-0.01676

0.00081

20

CSPA

0.5

0.5

0.11128

0.61058

-0.19494

0.05129

-0.00481

0.00017

21

CSPA

0.5

0.5

0.12346

0.50432

-0.13261

0.0402

-0.00448

0.00021

22

CSPA

0.5

0.5

0.09728

0.57515

-0.15977

0.04223

-0.00374

0.00012

23

CSPA

0.5

0.5

0.09455

0.61669

-0.22431

0.07407

-0.01002

0.00054

24

RCPA

0.5

0.5

0.16884

0.38783

-0.03679

0.01173

-0.00066

0.00002

25

RCPA

grv. hdwl

0.2

0.5

0.1301

0.43477

-0.07911

0.01764

-0.00114

0.00002

26

RCPA

grv. proj

0.2

0.5

0.09618

0.52593

-0.13504

0.03394

-0.00325

0.00013

EQ #'s: REFERENCE

12-23: Calculator Design Series (CDS) 4 for TI-59, FHWA, 1982, page 17

24-26: Calculator Design Series (CDS) 4 for TI-59, FHWA, 1982, page 24

12,16,20: Hydraulic Computer Program (HY) 2, FHWA, 1969, page 17

Table 6. Polynomial Coefficients - Concrete Open-Bottom Arch

Span:Rise Ratio Wingwall Angle (Inlet Configuration) KE SR A BS C DIP EE F Diagram/Notes
2:1

0 Degrees (Mitered to Conform to Slope)

0.7

0.0

0.0389106557

0.6044131889

-0.1966160961

0.0425827445

-0.0035136880

0.0001097816

2:1 Coefficients are used if the span:rise ratio is less than or equal to 3:1.

2:1

45 Degrees (45-degree Wingwall)

0.5

0.0

0.0580199163

0.5826504262

-0.1654982156

0.0337114383

-0.0026437555

0.0000796275

2:1 Coefficients are used if the span:rise ratio is less than or equal to 3:1.

2:1

90 Degrees (Square Edge with Headwall)

0.5

0.0

0.0747688320

0.5517030198

-0.1403253664

0.0281511418

-0.0021405250

0.0000632552

2:1 Coefficients are used if the span:rise ratio is less than or equal to 3:1.

4:1

0 Degrees (Mitered to Conform to Slope)

0.7

0.0

0.0557401882

0.4998819105

-0.1249164198

0.0219465031

-0.0015177347

0.0000404218

4:1 coefficients are used if the span:rise ratio is greater than 3:1

4:1

45 Degrees (45-degree Wingwall)

0.5

0.0

0.0465032346

0.5446293346

-0.1571341119

0.0312822438

-0.0024007467

0.0000704011

4:1 coefficients are used if the span:rise ratio is greater than 3:1

4:1

90 Degrees (Square Edge with Headwall)

0.5

0.0

0.0401619369

0.5774418238

-0.1693724912

0.0328323405

-0.0024131276

0.0000668323

4:1 coefficients are used if the span:rise ratio is greater than 3:1

References for Concrete Open-bottom Arch polynomial coefficients:

• Thiele, Elizabeth A. Culvert Hydraulics: Comparison of Current Computer Models. (pp. 121-126), Brigham Young University Master's Thesis (2007).
• Chase, Don. Hydraulic Characteristics of CON/SPAN Bridge Systems. Submitted Study and Report (1999)

Table 7. Polynomial Coefficients - South Dakota Concrete Box

Description KE SR A BS C DIP EE F Diagram/Notes
Sketch 1: 30 degree-flared wingwalls; top edge beveled at 45 degrees

0.5

0.5

0.0176998563

0.5354484847

-0.1197176702

0.0175902318

-0.0005722076

-0.0000080574

Sketch 2: 30 degree-flared wingwalls; top edge beveled at 45 degrees; 2, 3, and 4 multiple barrels

0.5

0.5

0.0506647261

0.5535393634

-0.1599374238

0.0339859269

-0.0027470036

0.0000851484

Sketch 3: 30 degree-flared wingwalls; top edge beveled at 45 degrees; 2:1 to 4:1 span-to-rise ratio

0.5

0.5

0.0518005829

0.5892384653

-0.1901266252

0.0412149379

-0.0034312198

0.0001083949

Sketch 4: 30 degree-flared wingwalls; top edge beveled at 45 degrees; 15 degrees skewed headwall with multiple barrels

0.5

0.5

0.2212801152

0.6022032341

-0.1672369732

0.0313391792

-0.0024440549

0.0000743575

Sketch 5: 30 degree-flared wingwalls; top edge beveled at 45 degrees; 30 degrees to 45 degrees skewed headwall with multiple barrels

0.5

0.5

0.2431604850

0.5407556631

-0.1267568901

0.0223638322

-0.0016523399

0.0000490932

Sketches 6 & 7: 0 degree-flared wingwalls (extended sides); square-edged at crown and 0 degree-flared wingwalls (extended sides); top edge beveled at 45 degrees; 0- and 6-inch corner fillets

0.5

0.5

0.0493946080

0.7138391179

-0.2354755894

0.0473247331

-0.0036154348

0.0001033337

Sketches 8 & 9: 0 degree-flared wingwalls (extended sides); top edge beveled at 45 degrees; 2, 3, and 4 multiple barrels and 0 degree-flared wingwalls (extended sides); top edge beveled at 45 degrees; 2:1 to 4:1 span-to-rise ratio

0.5

0.5

0.1013668008

0.6600937637

-0.2133066786

0.0437022641

-0.0035224589

0.0001078198

Sketches 10 & 11: 0 degree-flared wingwalls (extended sides); crown rounded at 8-inch radius; 0- and 6-inch corner fillets and 0 degree-flared wingwalls (extended sides); crown rounded at 8-inch radius; 12-inch corner fillets

0.5

0.5

0.0745605288

0.6533033536

-0.1899798824

0.0350021004

-0.0024571627

0.0000642284

Sketch 12: 0 degree-flared wingwalls (extended sides); crown rounded at 8-inch radius; 12-inch corner fillets; 2, 3, and 4 multiple barrels

0.5

0.5

0.1321993533

0.5024365440

-0.1073286526

0.0183092064

-0.0013702887

0.0000423592

Sketch 13: 0 degree-flared wingwalls (extended sides); crown rounded at 8-inch radius; 12-inch corner fillets; 2:1 to 4:1 span-to-rise ratio.

0.5

0.5

0.1212726739

0.6497418331

-0.1859782730

0.0336300433

-0.0024121680

0.0000655665

References for South Dakota Concrete Box polynomial coefficients:

Table 8. User Defined, Open Bottom Arch, Low-Profile Arch, High-Profile Arch, and Metal Box HW/D Values.

Q/A*D^.5 = 0.5 1 2 3 4 5 6 7 8 9
HY-8 Interpolation Coefficients Inlet Configuration KE SR A(1) A(2) A(3) A(4) A(5) A(6) A(7) A(8) A(9) A(10)

1

Thin Edge Projecting

0.9

0.5

0.31

0.48

0.81

1.11

1.42

1.84

2.39

3.03

3.71

4.26

2

Mitered to Conform to Slope

0.7

-0.7

0.34

0.49

0.77

1.04

1.45

1.91

2.46

3.06

3.69

4.34

3

0.5

0.5

0.31

0.46

0.73

0.96

1.26

1.59

2.01

2.51

3.08

3.64

4

Beveled Edge

0.2

0.5

0.31

0.44

0.69

0.89

1.16

1.49

1.81

2.23

2.68

3.18

Reference for User-defined interpolation coefficients: FHWA HDS-5, Appendix D, Chart 52B