HY-8: POLYNOMIAL COEFFICIENTS

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 Grooved End in Headwall 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 Square Projecting 0.2 0.5 0.167287 0.558766 -0.159813 0.0420069 -0.00369252 0.000125169
9 Square Edge with Headwall (Concrete/PVC/HDPE) 0.5 0.5 0.087483 0.706578 -0.253295 0.0667001 -0.00661651 0.000250619
10 End Section 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 20% Embedded, Square Headwall 0.55 0.5 0.0899367985347424 0.363046722229086 -0.0683746513605387 0.0109593856642167 -0.000706535544154146 0.0000189546410047092
5 40% Embedded, Square Headwall 0.55 0.5 0.074298531535586 0.4273662972292 -0.0849120530113796 0.0157965200237501 -0.00102651687866388 0.0000260155937601425
6 50% Embedded, Square Headwall 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 degree) Headwall, 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 (90 degree) Headwall, 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 1:1 Bevel Headwall 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 headwall 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 headwall 0.5 0.5 0.11128 0.61058 -0.19494 0.05129 -0.00481 0.00017
21 CSPA headwall 0.5 0.5 0.12346 0.50432 -0.13261 0.0402 -0.00448 0.00021
22 CSPA headwall 0.5 0.5 0.09728 0.57515 -0.15977 0.04223 -0.00374 0.00012
23 CSPA headwall 0.5 0.5 0.09455 0.61669 -0.22431 0.07407 -0.01002 0.00054
24 RCPA headwall 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 0DegreeWingwallConspanCulvertDiagram.png 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 45DegreeWingwallConspanCulvertDiagram.png 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 90DegreeWingwallConspanCulvertDiagram.png 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 0DegreeWingwallConspanCulvertDiagram.png 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 45DegreeWingwallConspanCulvertDiagram.png 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 90DegreeWingwallConspanCulvertDiagram.png 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 HY8SouthDakotaSketch1.png
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 HY8SouthDakotaSketch2.png
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 HY8SouthDakotaSketch3.png
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 HY8SouthDakotaSketch4.png
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 HY8SouthDakotaSketch5.png
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 HY8SouthDakotaSketch6.png HY8SouthDakotaSketch7.png
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 HY8SouthDakotaSketch8.png HY8SouthDakotaSketch9.png
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 HY8SouthDakotaSketch10.png HY8SouthDakotaSketch11.png
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 HY8SouthDakotaSketch12.png
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 HY8SouthDakotaSketch13.png

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 Square Edge with Headwall 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

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