HY8:Polynomial Coefficients
For circular, box, elliptical, pipe arch, and concrete open-bottom arch (commonly called CON/SPAN) culverts, polynomial coefficients, found in Tables 1-5, are utilized in the inlet control headwater computations. Tables 6-10 show the HW/D points A(1) through A(10) for interpolation. Each row of coefficients represents different entrance conditions for different culvert shapes.
Table 1. Polynomial Coefficients - Circular
| HY-8 Equation | Inlet Edge Condition | 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 |
sq. proj. |
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 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, 198O, page 60
1-10: Hydraulic Computer Program (HY) 1, FHWA, 1969, page 18
Table 2. Polynomial Coefficients - Box
| HY-8 Equation | Inlet Edge Condition | 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 3. Polynomial Coefficients - Ellipse
| HY-8 Equation | PIPE | Inlet Edge Condition | 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 |
0.5 |
-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 4. Polynomial Coefficients - Pipe Arch
| HY-8 Equation | PIPE | Inlet Edge Condition | 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 5. Polynomial Coefficients - Concrete Open-Bottom Arch
| Span:Rise Ratio | Wingwall Angle (Inlet Edge Condition) | KE | SR | A | BS | C | DIP | EE | F | Diagram/Notes |
|---|---|---|---|---|---|---|---|---|---|---|
| 2:1 |
0 Degrees (Mitered to Conform to Slope) |
0.7 |
0.0 |
0.0356 |
0.5701 |
-0.162 |
0.0322 |
-0.0023 |
0.00006 |
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 |
0.0393 |
0.5828 |
-0.1518 |
0.0284 |
-0.002 |
0.00005 |
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 |
0.0459 |
0.5762 |
-0.1436 |
0.0269 |
-0.0019 |
0.00005 |
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 |
0.0309 |
0.5211 |
-0.1258 |
0.0207 |
-0.0013 |
0.00003 |
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 |
0.0318 |
0.5446 |
-0.1467 |
0.0273 |
-0.0019 |
0.00005 |
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 |
0.0283 |
0.5739 |
-0.1584 |
0.0292 |
-0.002 |
0.00005 |
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 6. 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:
- Thiele, Elizabeth A. Culvert Hydraulics: Comparison of Current Computer Models. (pp. 121-126), Brigham Young University Master's Thesis (2007).
- Effects of Inlet Geometry on Hydraulic Performance of Box Culverts (FHWA Publication No. FHWA-HRT-06-138, October 2006)
Table 6. 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 Edge Condition | 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
