Newton-Raphson-Bisection Algorithm:

Step 1: Perform the following initial steps:

If \(f(x_{LOW}) > f(x_{HIGH})\) then switch \(x_{LOW}\) with \(x_{HIGH}\).

If \(x\) is outside \([x_{LOW},x_{HIGH}]\) then set \(x = \frac{x_{LOW} + x_{HIGH}}{2}\).

Set \(\Delta x = |x_{HIGH} - x_{LOW}|\).

Evaluate \(f(x)\) and its derivative \(f'(x)\).

Step 2: If \([(x-x_{HIGH})f'(x)-f(x)][(x-x_{LOW})f'(x)-f(x)] \geq 0\) or \(|2f(x)| > |\Delta x f'(x)|\) then update \(x\) as follows:

\(\Delta x = 0.5(x_{HIGH} - x_{LOW})\)

\(x = x_{LOW} + \Delta x\)

Otherwise take the Newton step:

\(\Delta x = \frac{f(x)}{f'(x)}\)

\(x = x - \Delta x\)

Step 3: If \(|\Delta x|\) is below the tolerance \(\varepsilon\) then stop with the current value of \(x\) as the solution.

Step 4: Evaluate \(f(x)\) and \(f'(x)\). If \(f(x) < 0\) then set \(x_{LOW} = x\). Otherwise set \(x_{HIGH} = x\).

Step 5: Return to Step 2.

Ridder's Root Finding Method

Ridder's method uses the following iterative procedure adapted from Press et al. (1992) to solve the equation \(f(x) = 0\) over the interval \(\left\lbrack x_{1},x_{2} \right\rbrack\) that is known to bracket the solution with a convergence tolerance of ε:

Ridder's Root Finding Algorithm:

Step 1: Let

\(f_1 = f(x_1)\)

\(f_2 = f(x_2)\)

\(x_3 = \frac{x_1 + x_2}{2}\)

Step 2: Set

\(f_3 = f(x_3)\)

\(x_4 = x_3 + \frac{(x_3 - x_1)\text{sgn}(f_1 - f_2)f_3}{\sqrt{f_3^2 - f_1 f_2}}\)

Step 3: If \(|x_4 - x_3| < \varepsilon\) then stop with solution \(x_3\).

Step 4: Set \(f_4 = f(x_4)\).

Step 5: If \(\text{sgn}(f_3) \neq \text{sgn}(f_4)\) then set

\(x_1 = x_3\)

\(f_1 = f_3\)

\(x_2 = x_4\)

\(f_2 = f_4\)

Otherwise if \(\text{sgn}(f_1) \neq \text{sgn}(f_4)\) then set

\(x_2 = x_4\)

\(f_2 = f_4\)

Otherwise if \(\text{sgn}(f_2) \neq \text{sgn}(f_4)\) then set

\(x_1 = x_4\)

\(f_1 = f_4\)

Step 6: Set \(x_3 = \frac{x_1 + x_2}{2}\).

Step 7: If \(|x_2 - x_1| \leq \varepsilon\) then stop with solution \(x_3\).

Step 8: Return to Step 2.

Section Properties of Circular Pipes

Table C‑1 Circular section properties as function of depth

Y/Y_full	A/A_full	W/W_max	R/R_full	Y/Y_full	A/A_full	W/W_max	R/R_full
0.00	0.00000	0.00000	0.01000	0.52	0.52550	0.99920	1.02400
0.02	0.00471	0.28000	0.05280	0.54	0.55093	0.99680	1.04800
0.04	0.01340	0.39190	0.10480	0.56	0.57630	0.99280	1.07000
0.06	0.024446	0.47500	0.15560	0.58	0.60135	0.98710	1.09120
0.08	0.03740	0.54260	0.20520	0.60	0.62640	0.97980	1.11000
0.10	0.05208	0.60000	0.25400	0.62	0.65126	0.97080	1.12720
0.12	0.06800	0.64990	0.30160	0.64	0.67580	0.96000	1.14400
0.14	0.08505	0.69400	0.34840	0.66	0.70015	0.94740	1.15960
0.16	0.10330	0.73320	0.39440	0.68	0.72410	0.93300	1.17400
0.18	0.12236	0.76840	0.43880	0.70	0.74764	0.91650	1.18480
0.20	0.14230	0.80000	0.48240	0.72	0.77080	0.89800	1.19400
0.22	0.16310	0.82850	0.52480	0.74	0.79335	0.87730	1.20240
0.24	0.18450	0.85420	0.56640	0.76	0.81540	0.85420	1.21000
0.26	0.20665	0.87730	0.60640	0.78	0.83690	0.82850	1.21480
0.28	0.22920	0.89800	0.64560	0.80	0.85760	0.80000	1.21700
0.30	0.25236	0.91650	0.68360	0.82	0.87764	0.76840	1.21720
0.32	0.27590	0.93300	0.72040	0.84	0.89670	0.73320	1.21500
0.34	0.29985	0.94740	0.75640	0.86	0.91495	0.69400	1.21040
0.36	0.32420	0.96000	0.79120	0.88	0.93200	0.64990	1.20300
0.38	0.34874	0.97080	0.82440	0.90	0.94792	0.60000	1.19200
0.40	0.37360	0.97980	0.85680	0.92	0.96260	0.54260	1.17800
0.42	0.39878	0.98710	0.88800	0.94	0.97555	0.47500	1.15840
0.44	0.42370	0.99280	0.91760	0.96	0.98660	0.39190	1.13200
0.46	0.44907	0.99680	0.94640	0.98	0.99516	0.28000	1.09400
0.48	0.47450	0.99920	0.97360	1.00	1.00000	0.00000	1.00000
0.50	0.50000	1.00000	1.00000

Table C‑2 Circular section properties as function of area

A/A_full	Y/Y_full	Ψ/Ψ_full	A/A_full	Y/Y_full	Ψ/Ψ_full
0.00	0.0000	0.00000	0.52	0.51572	0.52658
0.02	0.05236	0.00529	0.54	0.53146	0.55354
0.04	0.08369	0.01432	0.56	0.54723	0.58064
0.06	0.11025	0.02559	0.58	0.56305	0.60777
0.08	0.13423	0.03859	0.60	0.57892	0.63499
0.10	0.15643	0.05304	0.62	0.59487	0.66232
0.12	0.17755	0.06877	0.64	0.61093	0.68995
0.14	0.19772	0.08551	0.66	0.62710	0.71770
0.16	0.21704	0.10326	0.68	0.64342	0.74538
0.18	0.23581	0.12195	0.70	0.65991	0.77275
0.20	0.25412	0.14144	0.72	0.67659	0.79979
0.22	0.27194	0.16162	0.74	0.69350	0.82658
0.24	0.28948	0.18251	0.76	0.71068	0.85320
0.26	0.30653	0.20410	0.78	0.72816	0.87954
0.28	0.32349	0.22636	0.80	0.74602	0.90546
0.30	0.34017	0.24918	0.82	0.76424	0.93095
0.32	0.35666	0.27246	0.84	0.78297	0.95577
0.34	0.37298	0.29614	0.86	0.80235	0.97976
0.36	0.38915	0.32027	0.88	0.82240	1.00291
0.38	0.40521	0.34485	0.90	0.84353	1.02443
0.40	0.42117	0.36989	0.92	0.86563	1.04465
0.42	0.43704	0.39531	0.94	0.88970	1.06135
0.44	0.45284	0.42105	0.96	0.91444	1.08208
0.46	0.46858	0.44704	0.98	0.94749	1.07662
0.48	0.48430	0.47329	1.00	1.0000	1.00000
0.50	0.50000	0.49980

Section Properties of Elliptical Pipes

Table D‑1 Standard elliptical pipe sizes

Code	Minor Axis (in)	Major Axis (in)	A_full (ft²)	R_full (ft)
1	14	23	1.80	0.367
2	19	30	3.30	0.490
3	22	34	4.10	0.546
4	24	38	5.10	0.613
5	27	42	6.30	0.686
6	29	45	7.40	0.736
7	32	49	8.80	0.812
8	34	53	10.20	0.875
9	38	60	12.90	0.969
10	43	68	16.60	1.106
11	48	76	20.50	1.229
12	53	83	24.80	1.352
13	58	91	29.50	1.475
14	63	98	34.60	1.598
15	68	106	40.10	1.721
16	72	113	46.10	1.845
17	77	121	52.40	1.967
18	82	128	59.20	2.091
19	87	136	66.40	2.215
20	92	143	74.00	2.340
21	97	151	82.00	2.461
22	106	166	99.20	2.707
23	116	180	118.60	2.968

Note: The Minor Axis is the maximum width for a vertical ellipse and the full depth for a horizontal ellipse while the Major Axis is the maximum width for a horizontal ellipse and the full depth for a vertical ellipse.

Source: American Concrete Pipe Association (2011).

Table D‑2 Elliptical section properties as function of depth

Y/Y_full	A/A_full (Horizontal)	W/W_max (Horizontal)	R/R_full (Horizontal)	A/A_full (Vertical)	W/W_max (Vertical)	R/R_full (Vertical)
0.00	0.000	0.0000	0.0100	0.000	0.0000	0.0100
0.04	0.015	0.3919	0.0764	0.010	0.3919	0.1250
0.08	0.040	0.5426	0.1726	0.040	0.5426	0.2436
0.12	0.650	0.6499	0.2389	0.070	0.6499	0.3536
0.16	0.950	0.7332	0.3274	0.100	0.7332	0.4474
0.20	0.130	0.8000	0.4191	0.140	0.8000	0.5484
0.24	0.165	0.8542	0.5120	0.185	0.8542	0.6366
0.28	0.205	0.8980	0.5983	0.230	0.8980	0.7155
0.32	0.250	0.9330	0.6757	0.280	0.9330	0.7768
0.36	0.300	0.9600	0.7630	0.330	0.9600	0.8396
0.40	0.355	0.9798	0.8326	0.380	0.9798	0.8969
0.44	0.415	0.9928	0.9114	0.430	0.9928	0.9480
0.48	0.480	0.9992	0.9702	0.480	0.9992	0.9925
0.52	0.520	0.9992	1.030	0.520	0.9992	1.023
0.56	0.585	0.9928	1.091	0.570	0.9928	1.053
0.60	0.645	0.9798	1.146	0.620	0.9798	1.084
0.64	0.700	0.9600	1.185	0.670	0.9600	1.107
0.68	0.750	0.9330	1.225	0.720	0.9330	1.130
0.72	0.795	0.8980	1.257	0.770	0.8980	1.154
0.76	0.835	0.8542	1.274	0.815	0.8542	1.170
0.80	0.870	0.8000	1.290	0.860	0.8000	1.177
0.84	0.905	0.7332	1.282	0.900	0.7332	1.177
0.88	0.935	0.6499	1.274	0.930	0.6499	1.170
0.92	0.960	0.5426	1.257	0.960	0.5426	1.162
0.96	0.985	0.3919	1.185	0.990	0.3919	1.122
1.00	1.000	0.0000	1.000	1.000	0.0000	1.000

Section Properties of Arch Pipes

Table E‑1 Standard arch pipe sizes

Code	Material	Rise (Y_full) (in)	Span (W_max) (in)	A_full (in²)	R_full (in)
Concrete
1	Concrete	11	18	1.1	0.25
2	Concrete	13.5	22	1.65	0.30
3	Concrete	15.5	26	2.2	0.36
4	Concrete	18	28.5	2.8	0.45
5	Concrete	22.5	36.25	4.4	0.56
6	Concrete	26.625	43.75	6.4	0.68
7	Concrete	31.3125	51.125	8.8	0.80
8	Concrete	36	58.5	11.4	0.90
9	Concrete	40	65	14.3	1.01
10	Concrete	45	73	17.7	1.13
11	Concrete	54	88	25.6	1.35
12	Concrete	62	102	34.6	1.57
13	Concrete	72	115	44.5	1.77
14	Concrete	77.5	122	51.7	1.92
15	Concrete	87.125	138	66.0	2.17
16	Concrete	96.875	154	81.8	2.42
17	Concrete	106.5	168.75	99.1	2.65
Corrugated Steel, 2-2/3 x 1/2" Corrugation
18	Corrugated Steel	13	17	1.1	0.324
19	Corrugated Steel	15	21	1.6	0.374
20	Corrugated Steel	18	24	2.2	0.449
21	Corrugated Steel	20	28	2.9	0.499
22	Corrugated Steel	24	35	4.5	0.598
23	Corrugated Steel	29	42	6.5	0.723
24	Corrugated Steel	33	49	8.9	0.823
25	Corrugated Steel	38	57	11.6	0.947
26	Corrugated Steel	43	64	14.7	1.072
27	Corrugated Steel	47	71	18.1	1.171
28	Corrugated Steel	52	77	21.9	1.296
29	Corrugated Steel	57	83	26.0	1.421
Corrugated Steel, 3 x 1" Corrugation
30	Corrugated Steel	31	40	7.0	0.773
31	Corrugated Steel	36	46	9.4	0.773
32	Corrugated Steel	41	53	12.3	1.022
33	Corrugated Steel	46	60	15.6	1.147
34	Corrugated Steel	51	66	19.3	1.271
35	Corrugated Steel	55	73	23.2	1.371
36	Corrugated Steel	59	81	27.4	1.471
37	Corrugated Steel	63	87	32.1	1.570
38	Corrugated Steel	67	95	37.0	1.670
39	Corrugated Steel	71	103	42.4	1.770
40	Corrugated Steel	75	112	48.0	1.869
41	Corrugated Steel	79	117	54.2	1.969
42	Corrugated Steel	83	128	60.5	2.069
43	Corrugated Steel	87	137	67.4	2.168
44	Corrugated Steel	91	142	74.5	2.268
Structural Plate, 18" Corner Radius
45	Structural Plate	55	73	22	1.371
46	Structural Plate	57	76	24	1.421
47	Structural Plate	59	81	26	1.471
48	Structural Plate	61	84	28	1.520
49	Structural Plate	63	87	31	1.570
50	Structural Plate	65	92	33	1.620
51	Structural Plate	67	95	35	1.670
52	Structural Plate	69	98	38	1.720
53	Structural Plate	71	103	40	1.770
54	Structural Plate	73	106	43	1.820
55	Structural Plate	75	112	46	1.869
56	Structural Plate	77	114	49	1.919
57	Structural Plate	79	117	52	1.969
58	Structural Plate	81	123	55	2.019
59	Structural Plate	83	128	58	2.069
60	Structural Plate	85	131	61	2.119
61	Structural Plate	87	137	64	2.168
62	Structural Plate	89	139	67	2.218
63	Structural Plate	91	142	71	2.268
64	Structural Plate	93	148	74	2.318
65	Structural Plate	95	150	78	2.368
66	Structural Plate	97	152	81	2.418
67	Structural Plate	100	154	85	2.493
68	Structural Plate	101	161	89	2.517
69	Structural Plate	103	167	93	2.567
70	Structural Plate	105	169	97	2.617
71	Structural Plate	107	171	101	2.667
72	Structural Plate	109	178	105	2.717
73	Structural Plate	111	184	109	2.767
74	Structural Plate	113	186	113	2.817
75	Structural Plate	115	188	118	2.866
76	Structural Plate	118	190	122	2.941
77	Structural Plate	119	197	126	2.966
78	Structural Plate	121	199	131	3.016
Structural Plate, 31" Corner Radius
79	Structural Plate	112	159	97	2.792
80	Structural Plate	114	162	102	2.841
81	Structural Plate	116	168	105	2.891
82	Structural Plate	118	170	109	2.941
83	Structural Plate	120	173	114	2.991
84	Structural Plate	122	179	118	3.041
85	Structural Plate	124	184	123	3.091
86	Structural Plate	126	187	127	3.141
87	Structural Plate	128	190	132	3.190
88	Structural Plate	130	195	137	3.240
89	Structural Plate	132	198	142	3.290
90	Structural Plate	134	204	146	3.340
91	Structural Plate	136	206	151	3.390
92	Structural Plate	138	209	157	3.440
93	Structural Plate	140	215	161	3.490
94	Structural Plate	142	217	167	3.539
95	Structural Plate	144	223	172	3.589
96	Structural Plate	146	225	177	3.639
97	Structural Plate	148	231	182	3.689
98	Structural Plate	150	234	188	3.739
99	Structural Plate	152	236	194	3.789
100	Structural Plate	154	239	200	3.838
101	Structural Plate	156	245	205	3.888
102	Structural Plate	158	247	211	3.938

Source: American Iron and Steel Institute (1999).

Table E‑2 Arch pipe section properties as function of depth

Y/Y_full	A/A_full	W/W_max	R/R_full
0.00	0.000	0.0000	0.0100
0.04	0.020	0.6272	0.0983
0.08	0.060	0.8521	0.1965
0.12	0.100	0.9243	0.2948
0.16	0.140	0.9645	0.3940
0.20	0.190	0.9846	0.4962
0.24	0.240	0.9964	0.5911
0.28	0.290	0.9988	0.6796
0.32	0.340	0.9917	0.7615
0.36	0.390	0.9811	0.8364
0.40	0.440	0.9680	0.9044
0.44	0.490	0.9515	0.9640
0.48	0.540	0.9314	1.018
0.52	0.590	0.9101	1.065
0.56	0.640	0.8864	1.106
0.60	0.690	0.8592	1.142
0.64	0.735	0.8284	1.170
0.68	0.780	0.7917	1.192
0.72	0.820	0.7527	1.208
0.76	0.860	0.7065	1.217
0.80	0.895	0.6544	1.220
0.84	0.930	0.5953	1.213
0.88	0.960	0.5231	1.196
0.92	0.985	0.4355	1.168
0.96	0.995	0.3195	1.112
1.00	1.000	0.000	1.000

Section Properties of Masonry Sewers

Table F‑1 Area of masonry sewers as function of depth

Y/Y_full	Basket Handle	Egg	Horseshoe
0.00	0.0000	0.000	0.0000
0.04	0.0173	0.015	0.0181
0.08	0.0457	0.040	0.0508
0.12	0.0828	0.055	0.0908
0.16	0.1271	0.085	0.1326
0.20	0.1765	0.120	0.1757
0.24	0.2270	0.155	0.2201
0.28	0.2775	0.190	0.2655
0.32	0.3280	0.225	0.3118
0.36	0.3780	0.275	0.3587
0.40	0.4270	0.320	0.4064
0.44	0.4765	0.370	0.4542
0.48	0.5260	0.420	0.5023
0.52	0.5740	0.470	0.5506
0.56	0.6220	0.515	0.5987
0.60	0.6690	0.570	0.6462
0.64	0.7160	0.620	0.6931
0.68	0.7610	0.680	0.7387
0.72	0.8030	0.730	0.7829
0.76	0.8390	0.780	0.8253
0.80	0.8770	0.835	0.8652
0.84	0.9110	0.885	0.9022
0.88	0.9410	0.925	0.9356
0.92	0.9680	0.955	0.9645
0.96	0.9880	0.980	0.9873
1.00	1.0000	1.000	1.0000

Table F‑2 Width of masonry sewers as function of depth - I

Y/Y_full	Basket Handle	Egg	Horseshoe
0.00	0.000	0.000	0.0000
0.04	0.490	0.298	0.5878
0.08	0.667	0.433	0.8772
0.12	0.820	0.508	0.8900
0.16	0.930	0.582	0.9028
0.20	1.000	0.642	0.9156
0.24	1.000	0.696	0.9284
0.28	1.000	0.746	0.9412
0.32	0.997	0.791	0.9540
0.36	0.994	0.836	0.9668
0.40	0.988	0.866	0.9798
0.44	0.982	0.896	0.9928
0.48	0.967	0.926	0.9992
0.52	0.948	0.956	0.9992
0.56	0.928	0.970	0.9928
0.60	0.904	0.985	0.9798
0.64	0.874	1.000	0.9600
0.68	0.842	0.985	0.9330
0.72	0.798	0.970	0.8980
0.76	0.750	0.940	0.8542
0.80	0.697	0.896	0.8000
0.84	0.637	0.836	0.7332
0.88	0.567	0.764	0.6499
0.92	0.467	0.642	0.5426
0.96	0.342	0.310	0.3919
1.00	0.000	0.000	0.0000

Table F‑3 Width of masonry sewers as function of depth - II

Y/Y_full	Gothic	Catenary	Semi-Elliptical	Semi-Circular
0.00	0.000	0.0000	0.00	0.0000
0.05	0.286	0.6667	0.70	0.5488
0.10	0.643	0.8222	0.98	0.8537
0.15	0.762	0.9111	1.00	1.0000
0.20	0.833	0.9778	1.00	1.0000
0.25	0.905	1.0000	1.00	0.9939
0.30	0.952	1.0000	0.99	0.9878
0.35	0.976	0.9889	0.98	0.9756
0.40	0.976	0.9778	0.96	0.9634
0.45	1.000	0.9556	0.94	0.9451
0.50	1.000	0.9333	0.91	0.9207
0.55	0.976	0.8889	0.88	0.8902
0.60	0.976	0.8444	0.84	0.8537
0.65	0.952	0.8000	0.80	0.8171
0.70	0.905	0.7556	0.75	0.7683
0.75	0.833	0.7000	0.70	0.7073
0.80	0.762	0.6333	0.64	0.6463
0.85	0.667	0.5556	0.56	0.5732
0.90	0.524	0.4444	0.46	0.4756
0.95	0.357	0.3333	0.34	0.3354
1.00	0.000	0.0000	0.00	0.0000

Table F‑4 Hydraulic radius of masonry sewers as function of depth

Y/Y_full	R/R_full (Basket Handle)	R/R_full (Egg)	R/R_full (Horseshoe)
0.00	0.010	0.010	0.0100
0.04	0.0952	0.097	0.1040
0.08	0.189	0.216	0.2065
0.12	0.273	0.302	0.3243
0.16	0.369	0.386	0.4322
0.20	0.463	0.465	0.5284
0.24	0.560	0.536	0.6147
0.28	0.653	0.611	0.6927
0.32	0.743	0.676	0.7636
0.36	0.822	0.735	0.8268
0.40	0.883	0.791	0.8873
0.44	0.949	0.854	0.9417
0.48	0.999	0.904	0.9905
0.52	1.055	0.941	1.036
0.56	1.095	1.008	1.077
0.60	1.141	1.045	1.113
0.64	1.161	1.076	1.143
0.68	1.188	1.115	1.169
0.72	1.206	1.146	1.189
0.76	1.206	1.162	1.202
0.80	1.206	1.186	1.208
0.84	1.205	1.193	1.206
0.88	1.196	1.186	1.195
0.92	1.168	1.162	1.170
0.96	1.127	1.107	1.126
1.00	1.000	1.000	1.000

Table F‑5 Depth of masonry sewers as function of area - I

A/A_full	Y/Y_full (Basket Handle)	Y/Y_full (Egg)	Y/Y_full (Horseshoe)
0.00	0.00000	0.00000	0.00000
0.02	0.04112	0.04912	0.04146
0.04	0.07380	0.08101	0.07033
0.06	0.10000	0.11128	0.09098
0.08	0.12236	0.14161	0.10962
0.10	0.14141	0.16622	0.12921
0.12	0.15857	0.18811	0.14813
0.14	0.17462	0.21356	0.16701
0.16	0.18946	0.23742	0.18565
0.18	0.20315	0.25742	0.20401
0.20	0.21557	0.27742	0.22211
0.22	0.22833	0.29741	0.23998
0.24	0.24230	0.31742	0.25769
0.26	0.25945	0.33742	0.27524
0.28	0.27936	0.35747	0.29265
0.30	0.30000	0.37364	0.30990
0.32	0.32040	0.40000	0.32704
0.34	0.34034	0.41697	0.34406
0.36	0.35892	0.43372	0.36101
0.38	0.37595	0.45000	0.37790
0.40	0.39214	0.46374	0.39471
0.42	0.40802	0.47747	0.41147
0.44	0.42372	0.49209	0.42818
0.46	0.43894	0.50989	0.44484
0.48	0.45315	0.53015	0.46147
0.50	0.46557	0.55000	0.47807
0.52	0.47833	0.56429	0.49468
0.54	0.49230	0.57675	0.51134
0.56	0.50945	0.58834	0.52803
0.58	0.52936	0.60000	0.54474
0.60	0.55000	0.61441	0.56138
0.62	0.57000	0.62967	0.57804
0.64	0.59000	0.64582	0.59478
0.66	0.61023	0.66368	0.61171
0.68	0.63045	0.68209	0.62881
0.70	0.65000	0.70000	0.64609
0.72	0.66756	0.71463	0.66350
0.74	0.68413	0.72807	0.68111
0.76	0.70000	0.74074	0.69901
0.78	0.71481	0.75296	0.71722
0.80	0.72984	0.76500	0.73583
0.82	0.74579	0.77784	0.75490
0.84	0.76417	0.79212	0.77447
0.86	0.78422	0.80945	0.79471
0.88	0.80477	0.82936	0.81564
0.90	0.82532	0.85000	0.83759
0.92	0.85000	0.86731	0.86067
0.94	0.88277	0.88769	0.88557
0.96	0.91500	0.91400	0.91159
0.98	0.95000	0.95000	0.94520
1.00	1.00000	1.00000	1.00000

Table F‑6 Depth of masonry sewers as function of area - II

A/A_full	Y/Y_full (Catenary)	Y/Y_full (Gothic)	Y/Y_full (Semi-Circular)	Y/Y_full (Semi-Elliptical)
0.00	0.00000	0.00000	0.00000	0.00000
0.02	0.02974	0.04522	0.04102	0.03075
0.04	0.06439	0.07825	0.07407	0.05137
0.06	0.08433	0.10646	0.10000	0.07032
0.08	0.10549	0.12645	0.11769	0.09000
0.10	0.12064	0.14645	0.13037	0.11323
0.12	0.13952	0.16787	0.14036	0.13037
0.14	0.15560	0.18641	0.15000	0.14519
0.16	0.17032	0.20129	0.16546	0.15968
0.18	0.18512	0.22425	0.18213	0.18459
0.20	0.20057	0.24129	0.20000	0.19531
0.22	0.21995	0.25624	0.22018	0.21354
0.24	0.24011	0.27344	0.24030	0.22694
0.26	0.25892	0.29097	0.25788	0.23947
0.28	0.27595	0.30529	0.27216	0.25296
0.30	0.29214	0.32607	0.28500	0.26500
0.32	0.30802	0.33755	0.29704	0.27784
0.34	0.32372	0.35073	0.30892	0.29212
0.36	0.33894	0.36447	0.32128	0.30970
0.38	0.35315	0.37558	0.33476	0.32982
0.40	0.36557	0.40000	0.35000	0.35000
0.42	0.37833	0.41810	0.36927	0.36738
0.44	0.39230	0.43648	0.38963	0.38390
0.46	0.40970	0.45374	0.41023	0.40000
0.48	0.42982	0.46805	0.43045	0.41667
0.50	0.45000	0.48195	0.45000	0.43333
0.52	0.46769	0.49626	0.46769	0.45000
0.54	0.48431	0.51352	0.48431	0.46697
0.56	0.50000	0.53190	0.50000	0.48372
0.58	0.51466	0.55000	0.51443	0.50000
0.60	0.52886	0.56416	0.52851	0.51374
0.62	0.54292	0.57787	0.54271	0.52747
0.64	0.55729	0.59224	0.55774	0.54209
0.66	0.57223	0.60950	0.57388	0.55950
0.68	0.58780	0.62941	0.59101	0.57941
0.70	0.60428	0.65000	0.60989	0.60000
0.72	0.62197	0.67064	0.63005	0.62000
0.74	0.64047	0.69055	0.65000	0.64000
0.76	0.65980	0.70721	0.66682	0.66000
0.78	0.67976	0.72031	0.68318	0.68000
0.80	0.70000	0.73286	0.70000	0.70000
0.82	0.71731	0.74632	0.71675	0.71843
0.84	0.73769	0.76432	0.73744	0.73865
0.86	0.76651	0.78448	0.76651	0.76365
0.88	0.80000	0.80421	0.80000	0.79260
0.90	0.82090	0.82199	0.82090	0.82088
0.92	0.84311	0.84363	0.84311	0.85000
0.94	0.87978	0.87423	0.87978	0.88341
0.96	0.91576	0.90617	0.91576	0.90998
0.98	0.95000	0.93827	0.95000	0.93871
1.00	1.00000	1.00000	1.00000	1.00000

Table F‑7 Section factor for masonry sewers as function of area - I

A/A_full	Ψ/Ψ_full (Basket Handle)	Ψ/Ψ_full (Egg)	Ψ/Ψ_full (Horseshoe)
0.00	0.00000	0.00000	0.00000
0.02	0.00758	0.00295	0.00467
0.04	0.01812	0.01331	0.01237
0.06	0.03000	0.02629	0.02268
0.08	0.03966	0.04000	0.03515
0.10	0.04957	0.05657	0.04943
0.12	0.06230	0.07500	0.06525
0.14	0.07849	0.09432	0.08212
0.16	0.09618	0.11473	0.10005
0.18	0.11416	0.13657	0.11891
0.20	0.13094	0.15894	0.13856
0.22	0.14808	0.18030	0.15896
0.24	0.16583	0.20036	0.18004
0.26	0.18381	0.22000	0.20172
0.28	0.20294	0.23919	0.22397
0.30	0.22500	0.25896	0.24677
0.32	0.25470	0.28000	0.27006
0.34	0.28532	0.30504	0.29380
0.36	0.31006	0.33082	0.31790
0.38	0.32804	0.35551	0.34237
0.40	0.34555	0.37692	0.36720
0.42	0.36944	0.39809	0.39239
0.44	0.40032	0.42000	0.41792
0.46	0.43203	0.44625	0.44374
0.48	0.46004	0.47321	0.46984
0.50	0.47849	0.50000	0.49619
0.52	0.49591	0.52255	0.52276
0.54	0.51454	0.54481	0.54950
0.56	0.53810	0.56785	0.57640
0.58	0.56711	0.59466	0.60345
0.60	0.60000	0.62485	0.63065
0.62	0.64092	0.65518	0.65795
0.64	0.68136	0.68181	0.68531
0.66	0.71259	0.70415	0.71271
0.68	0.73438	0.72585	0.74009
0.70	0.75500	0.74819	0.76738
0.72	0.78625	0.77482	0.79451
0.74	0.81880	0.80515	0.82144
0.76	0.85000	0.83534	0.84814
0.78	0.86790	0.86193	0.87450
0.80	0.88483	0.88465	0.90057
0.82	0.90431	0.90690	0.92652
0.84	0.93690	0.93000	0.95244
0.86	0.97388	0.95866	0.97724
0.88	1.00747	0.98673	0.99988
0.90	1.03300	1.01238	1.02048
0.92	1.05000	1.03396	1.03989
0.94	1.05464	1.05000	1.05698
0.96	1.06078	1.06517	1.07694
0.98	1.05500	1.05380	1.07562
1.00	1.00000	1.00000	1.00000

Table F‑8 Section factor for masonry sewers as function of area - II

A/A_full	Ψ/Ψ_full (Catenary)	Ψ/Ψ_full (Gothic)	Ψ/Ψ_full (Semi-Circular)	Ψ/Ψ_full (Semi-Elliptical)
0.00	0.00000	0.00000	0.00000	0.00000
0.02	0.00605	0.00500	0.00757	0.00438
0.04	0.01455	0.01740	0.01815	0.01227
0.06	0.02540	0.03098	0.03000	0.02312
0.08	0.03863	0.04272	0.03580	0.03638
0.10	0.05430	0.05500	0.04037	0.05145
0.12	0.07127	0.06980	0.04601	0.06783
0.14	0.08778	0.08620	0.05500	0.08500
0.16	0.10372	0.10461	0.07475	0.10093
0.18	0.12081	0.12463	0.09834	0.11752
0.20	0.14082	0.14500	0.12500	0.13530
0.22	0.16375	0.16309	0.15570	0.15626
0.24	0.18779	0.18118	0.18588	0.17917
0.26	0.21157	0.20000	0.20883	0.20296
0.28	0.23478	0.22181	0.22300	0.22654
0.30	0.25818	0.24487	0.23472	0.24962
0.32	0.28244	0.26888	0.24667	0.27269
0.34	0.30741	0.29380	0.26758	0.29568
0.36	0.33204	0.31901	0.29346	0.31848
0.38	0.35505	0.34389	0.32124	0.34152
0.40	0.37465	0.36564	0.35000	0.36500
0.42	0.39404	0.38612	0.37720	0.38941
0.44	0.41426	0.40720	0.40540	0.41442
0.46	0.43804	0.43000	0.43541	0.44000
0.48	0.46531	0.45868	0.46722	0.46636
0.50	0.49357	0.48895	0.50000	0.49309
0.52	0.52187	0.52000	0.53532	0.52000
0.54	0.54925	0.55032	0.56935	0.54628
0.56	0.57647	0.58040	0.60000	0.57285
0.58	0.60321	0.61000	0.61544	0.60000
0.60	0.62964	0.63762	0.62811	0.62949
0.62	0.65639	0.66505	0.64170	0.65877
0.64	0.68472	0.69290	0.66598	0.68624
0.66	0.71425	0.72342	0.70010	0.71017
0.68	0.74303	0.75467	0.73413	0.73304
0.70	0.76827	0.78500	0.76068	0.75578
0.72	0.79168	0.81165	0.78027	0.77925
0.74	0.81500	0.83654	0.80000	0.80368
0.76	0.84094	0.86000	0.82891	0.83114
0.78	0.86707	0.88253	0.85964	0.85950
0.80	0.89213	0.90414	0.89000	0.88592
0.82	0.91607	0.92500	0.91270	0.90848
0.84	0.94000	0.94486	0.93664	0.93000
0.86	0.96604	0.96475	0.96677	0.95292
0.88	0.99000	0.98567	1.00000	0.97481
0.90	1.00714	1.00833	1.02661	0.99374
0.92	1.02158	1.03000	1.04631	1.01084
0.94	1.03814	1.05360	1.05726	1.02858
0.96	1.05000	1.06500	1.06637	1.04543
0.98	1.05000	1.05500	1.06000	1.05000
1.00	1.00000	1.00000	1.00000	1.00000

Manning's Roughness Coefficients

Table G‑1 Manning's roughness coefficient n for open channels

Type of Channel and Description	Minimum	Normal	Maximum
1. Natural streams - minor streams (top width at flood stage < 100 ft)
a. clean, straight, full stage, no rifts or deep pools	0.025	0.030	0.033
b. same as above, but more stones and weeds	0.030	0.035	0.040
c. clean, winding, some pools and shoals	0.033	0.040	0.045
d. same as above, but some weeds and stones	0.035	0.045	0.050
e. same as above, lower stages, more ineffective slopes and sections	0.040	0.048	0.055
f. same as "d" with more stones	0.045	0.050	0.060
g. sluggish reaches, weedy, deep pools	0.050	0.070	0.080
h. very weedy reaches, deep pools, or floodways with heavy stand of timber and underbrush	0.075	0.100	0.150
2. Mountain streams, no vegetation in channel, banks usually steep, trees and brush along banks submerged at high stages
a. bottom: gravels, cobbles, and few boulders	0.030	0.040	0.050
b. bottom: cobbles with large boulders	0.040	0.050	0.070
3. Floodplains
a. Pasture, no brush
1. short grass	0.025	0.030	0.035
2. high grass	0.030	0.035	0.050
b. Cultivated areas
1. no crop	0.020	0.030	0.040
2. mature row crops	0.025	0.035	0.045
3. mature field crops	0.030	0.040	0.050
c. Brush
1. scattered brush, heavy weeds	0.035	0.050	0.070
2. light brush and trees, in winter	0.035	0.050	0.060
3. light brush and trees, in summer	0.040	0.060	0.080
4. medium to dense brush, in winter	0.045	0.070	0.110
5. medium to dense brush, in summer	0.070	0.100	0.160
d. Trees
1. dense willows, summer, straight	0.110	0.150	0.200
2. cleared land with tree stumps, no sprouts	0.030	0.040	0.050
3. same as above, but with heavy growth of sprouts	0.050	0.060	0.080
4. heavy stand of timber, a few down trees, little undergrowth, flood stage below branches	0.080	0.100	0.120
5. same as 4. with flood stage reaching branches	0.100	0.120	0.160
4. Excavated or Dredged Channels
a. Earth, straight, and uniform
1. clean, recently completed	0.016	0.018	0.020
2. clean, after weathering	0.018	0.022	0.025
3. gravel, uniform section, clean	0.022	0.025	0.030
4. with short grass, few weeds	0.022	0.027	0.033
b. Earth winding and sluggish
1. no vegetation	0.023	0.025	0.030
2. grass, some weeds	0.025	0.030	0.033
3. dense weeds or aquatic plants in deep channels	0.030	0.035	0.040
4. earth bottom and rubble sides	0.028	0.030	0.035
5. stony bottom and weedy banks	0.025	0.035	0.040
6. cobble bottom and clean sides	0.030	0.040	0.050
c. Dragline-excavated or dredged
1. no vegetation	0.025	0.028	0.033
2. light brush on banks	0.035	0.050	0.060
d. Rock cuts
1. smooth and uniform	0.025	0.035	0.040
2. jagged and irregular	0.035	0.040	0.050
e. Channels not maintained, weeds and brush uncut
1. dense weeds, high as flow depth	0.050	0.080	0.120
2. clean bottom, brush on sides	0.040	0.050	0.080
3. same as above, highest stage of flow	0.045	0.070	0.110
4. dense brush, high stage	0.080	0.100	0.140
5. Lined or Constructed Channels
a. Cement
1. neat surface	0.010	0.011	0.013
2. mortar	0.011	0.013	0.015
b. Wood
1. planed, untreated	0.010	0.012	0.014
2. planed, creosoted	0.011	0.012	0.015
3. unplaned	0.011	0.013	0.015
4. plank with battens	0.012	0.015	0.018
5. lined with roofing paper	0.010	0.014	0.017
c. Concrete
1. trowel finish	0.011	0.013	0.015
2. float finish	0.013	0.015	0.016
3. finished, with gravel on bottom	0.015	0.017	0.020
4. unfinished	0.014	0.017	0.020
5. gunite, good section	0.016	0.019	0.023
6. gunite, wavy section	0.018	0.022	0.025
7. on good excavated rock	0.017	0.020
8. on irregular excavated rock	0.022	0.027
d. Concrete bottom float finish with sides of:
1. dressed stone in mortar	0.015	0.017	0.020
2. random stone in mortar	0.017	0.020	0.024
3. cement rubble masonry, plastered	0.016	0.020	0.024
4. cement rubble masonry	0.020	0.025	0.030
5. dry rubble or riprap	0.020	0.030	0.035
e. Gravel bottom with sides of:
1. formed concrete	0.017	0.020	0.025
2. random stone mortar	0.020	0.023	0.026
3. dry rubble or riprap	0.023	0.033	0.036
f. Brick
1. glazed	0.011	0.013	0.015
2. in cement mortar	0.012	0.015	0.018
g. Masonry
1. cemented rubble	0.017	0.025	0.030
2. dry rubble	0.023	0.032	0.035
h. Dressed ashlar/stone paving	0.013	0.015	0.017
i. Asphalt
1. smooth	0.013	0.013
2. rough	0.016	0.016
j. Vegetal lining	0.030		0.500

Source: Chow, 1959.

Table G‑2 Manning's roughness coefficient n for closed conduits

Type of Conduit and Description	Minimum	Normal	Maximum
1. Brass, smooth:	0.009	0.010	0.013
2. Steel:
Lockbar and welded	0.010	0.012	0.014
Riveted and spiral	0.013	0.016	0.017
3. Cast Iron:
Coated	0.010	0.013	0.014
Uncoated	0.011	0.014	0.016
4. Wrought Iron:
Black	0.012	0.014	0.015
Galvanized	0.013	0.016	0.017
5. Corrugated Metal:
Subdrain	0.017	0.019	0.021
Stormdrain	0.021	0.024	0.030
6. Cement:
Neat Surface	0.010	0.011	0.013
Mortar	0.011	0.013	0.015
7. Concrete:
Culvert, straight and free of debris	0.010	0.011	0.013
Culvert with bends, connections, and some debris	0.011	0.013	0.014
Finished	0.011	0.012	0.014
Sewer with manholes, inlet, etc., straight	0.013	0.015	0.017
Unfinished, steel form	0.012	0.013	0.014
Unfinished, smooth wood form	0.012	0.014	0.016
Unfinished, rough wood form	0.015	0.017	0.020
8. Wood:
Stave	0.010	0.012	0.014
Laminated, treated	0.015	0.017	0.020
9. Clay:
Common drainage tile	0.011	0.013	0.017
Vitrified sewer	0.011	0.014	0.017
Vitrified sewer with manholes, inlet, etc.	0.013	0.015	0.017
Vitrified subdrain with open joint	0.014	0.016	0.018
10. Brickwork:
Glazed	0.011	0.013	0.015
Lined with cement mortar	0.012	0.015	0.017
Sanitary sewers coated with sewage slime with bends and connections	0.012	0.013	0.016
Paved invert, sewer, smooth bottom	0.016	0.019	0.020
Rubble masonry, cemented	0.018	0.025	0.030

Source: Chow, 1959.

Table G‑3 Manning's roughness coefficient n for corrugated steel pipe

Type of Pipe, Diameter and Corrugation Dimension	n
1. Annular 2.67 x 1/2 inch (all diameters)	0.024
2. Helical 1.50 x 1/4 inch
8" diameter	0.012
10" diameter	0.014
3. Helical 2.67 x 1/2 inch
12" diameter	0.011
18" diameter	0.014
24" diameter	0.016
36" diameter	0.019
48" diameter	0.020
60" diameter	0.021
4. Annular 3x1 inch (all diameters)	0.027
5. Helical 3x1 inch
48" diameter	0.023
54" diameter	0.023
60" diameter	0.024
66" diameter	0.025
72" diameter	0.026
78" diameter and larger	0.027
6. Corrugations 6x2 inches
60" diameter	0.033
72" diameter	0.032
120" diameter	0.030
180" diameter	0.028

Source: American Iron and Steel Institute, 1999.

Culvert Coefficients

Table H‑1 Culvert codes

Culvert Shape and Material	Inlet Configuration	Code
Circular Concrete	Square edge with headwall	1
	Groove end with headwall	2
	Groove end projecting	3
Circular Corrugated Metal Pipe	Headwall	4
	Mitered to slope	5
	Projecting	6
Circular Pipe, Beveled Ring Entrance	45 deg. bevels	7
	33.7 deg. bevels	8
Rectangular Box; Flared Wingwalls	30-75 deg. wingwall flares	9
	90 or 15 deg. wingwall flares	10
	0 deg. wingwall flares (straight sides)	11
Rectangular Box; Flared Wingwalls and Top Edge Bevel	45 deg flare; 0.43D top edge bevel	12
	18-33.7 deg. flare; 0.083D top edge bevel	13
Rectangular Box, 90-deg Headwall, Chamfered / Beveled Inlet Edges	Chamfered 3/4-in.	14
	Beveled 1/2-in/ft at 45 deg (1:1)	15
	Beveled 1-in/ft at 33.7 deg (1:1.5)	16
Rectangular Box, Skewed Headwall, Chamfered / Beveled Inlet Edges	3/4" chamfered edge, 45 deg skewed headwall	17
	3/4" chamfered edge, 30 deg skewed headwall	18
	3/4" chamfered edge, 15 deg skewed headwall	19
	45 deg beveled edge, 10-45 deg skewed headwall	20
Rectangular Box, Non-offset Flared Wingwalls, 3/4" Chamfer at Top of Inlet	45 deg (1:1) wingwall flare	21
	8.4 deg (3:1) wingwall flare	22
	18.4 deg (3:1) wingwall flare, 30 deg inlet skew	23
Rectangular Box, Offset Flared Wingwalls, Beveled Edge at Inlet Top	45 deg (1:1) flare, 0.042D top edge bevel	24
	33.7 deg (1.5:1) flare, 0.083D top edge bevel	25
	18.4 deg (3:1) flare, 0.083D top edge bevel	26
Corrugated Metal Box	90 deg headwall	27
	Thick wall projecting	28
	Thin wall projecting	29
Horizontal Ellipse Concrete	Square edge with headwall	30
	Grooved end with headwall	31
	Grooved end projecting	32
Vertical Ellipse Concrete	Square edge with headwall	33
	Grooved end with headwall	34
	Grooved end projecting	35
Pipe Arch, 18" Corner Radius, Corrugated Metal	90 deg headwall	36
	Mitered to slope	37
	Projecting	38
Pipe Arch, 18" Corner Radius, Corrugated Metal	Projecting	39
	No bevels	40
	33.7 deg bevels	41
Pipe Arch, 31" Corner Radius, Corrugated Metal	Projecting	42
	No bevels	43
	33.7 deg. bevels	44
Arch, Corrugated Metal	90 deg headwall	45
	Mitered to slope	46
	Thin wall projecting	47
Circular Culvert	Smooth tapered inlet throat	48
	Rough tapered inlet throat	49
Elliptical Inlet Face	Tapered inlet, beveled edges	50
	Tapered inlet, square edges	51
	Tapered inlet, thin edge projecting	52
Rectangular	Tapered inlet throat	53
Rectangular Concrete	Side tapered, less favorable edges	54
	Side tapered, more favorable edges	55
	Slope tapered, less favorable edges	56
	Slope tapered, more favorable edges	57

Table H‑2 Culvert coefficients

Culvert Code	Equation Form	Unsubmerged K	Unsubmerged M	Submerged c	Submerged y
1	1	0.0098	2.00	0.0398	0.67
2	1	0.0018	2.00	0.0292	0.74
3	1	0.0045	2.00	0.0317	0.69
4	1	0.0078	2.00	0.0379	0.69
5	1	0.0210	1.33	0.0463	0.75
6	1	0.0340	1.50	0.0553	0.54
7	1	0.0018	2.50	0.0300	0.74
8	1	0.0018	2.50	0.0243	0.83
9	1	0.026	1.0	0.0347	0.81
10	1	0.061	0.75	0.0400	0.80
11	1	0.061	0.75	0.0423	0.82
12	2	0.510	0.667	0.0309	0.80
13	2	0.486	0.667	0.0249	0.83
14	2	0.515	0.667	0.0375	0.79
15	2	0.495	0.667	0.0314	0.82
16	2	0.486	0.667	0.0252	0.865
17	2	0.545	0.667	0.04505	0.73
18	2	0.533	0.667	0.0425	0.705
19	2	0.522	0.667	0.0402	0.68
20	2	0.498	0.667	0.0327	0.75
21	2	0.497	0.667	0.0339	0.803
22	2	0.493	0.667	0.0361	0.806
23	2	0.495	0.667	0.0386	0.71
24	2	0.497	0.667	0.0302	0.835
25	2	0.495	0.667	0.0252	0.881
26	2	0.493	0.667	0.0227	0.887
27	1	0.0083	2.00	0.0379	0.69
28	1	0.0145	1.75	0.0419	0.64
29	1	0.0340	1.50	0.0496	0.57
30	1	0.0100	2.00	0.0398	0.67
31	1	0.0018	2.50	0.0292	0.74
32	1	0.0045	2.00	0.0317	0.69
33	1	0.0100	2.00	0.0398	0.67
34	1	0.0018	2.50	0.0292	0.74
35	1	0.0095	2.00	0.0317	0.69
36	1	0.0083	2.00	0.0379	0.69
37	1	0.0300	1.00	0.0463	0.75
38	1	0.0340	1.50	0.0496	0.57
39	1	0.0300	1.50	0.0496	0.57
40	1	0.0088	2.00	0.0368	0.68
41	1	0.0030	2.00	0.0269	0.77
42	1	0.0300	1.50	0.0496	0.57
43	1	0.0088	2.00	0.0368	0.68
44	1	0.0030	2.00	0.0269	0.77
45	1	0.0083	2.00	0.0379	0.69
46	1	0.0300	1.00	0.0473	0.75
47	1	0.0340	1.50	0.0496	0.57
48	2	0.534	0.555	0.0196	0.90
49	2	0.519	0.640	0.0210	0.90
50	2	0.536	0.622	0.0368	0.83
51	2	0.5035	0.719	0.0478	0.80
52	2	0.547	0.800	0.0598	0.75
53	2	0.475	0.667	0.0179	0.97
54	2	0.560	0.667	0.0446	0.85
55	2	0.560	0.667	0.0378	0.87
56	2	0.500	0.667	0.0446	0.65
57	2	0.500	0.667	0.0378	0.71

Table of Contents

Newton-Raphson-Bisection Root Finding Method