--- title: Benchmark of Equations for Pipe and Box Culvert description: "Some of the calculation in culvert require you to refer to complicated graphs in order to obtain the output. Graphs used in the design can be referred in the previous **[**blog**](https://mes100.com/blog/graphs-and-equations-used-in-culvert-design-nomographs-part-full-flow-critical-depth-full-flow/)** post. Therefore, to reduce the difficulties, we provide you…" url: "https://mes100.com/docs/benchmark-of-equations-for-pipe-and-box-culvert/" updated: "2022-12-05" category: Benchmark / Drain --- # Benchmark of Equations for Pipe and Box Culvert Some of the calculation in culvert require you to refer to complicated graphs in order to obtain the output. Graphs used in the design can be referred in the previous **[**blog**](https://mes100.com/blog/graphs-and-equations-used-in-culvert-design-nomographs-part-full-flow-critical-depth-full-flow/)** post. Therefore, to reduce the difficulties, we provide you the equations and examples to benchmark with the equation. # **Inlet Control Nomograph for Pipe Culvert and Box Culvert** Based on Design Chart 18.A2 and Design Chart 18.A3 Derived equation to obtain the result of inlet control can be referred as follows; $$ \style{font-family:Arial}{{\mathrm{HW}}_{\mathrm i}\;=\;\mathrm c{(\frac{\mathrm{Ku}.\mathrm Q}{\mathrm A.\mathrm D^{0.5}})}^2\;+\;\mathrm Y\;+\;\mathrm{Ks}.\mathrm S} $$ HWi = Headwater depth (m) D = Diameter of culvert barrel (m) c = Constant from Table 1.1, Table 1.2, Table 1.3 Ku = Unit Conversion, 1.811 SI unit Q = Discharge (m3/s) A = Full cross sectional area of culvert barel (m2) Y = Constant from Table 1.1, Table 1.2, Table 1.3 Ks = Slope correction, -0.5 S = Culvert barrel slope (m/m) The examples to benchmark with the equation are as follows. - [**Design Chart 18.A2 for Pipe Culvert**](https://docs.google.com/spreadsheets/d/1Vjij2eh9WiW5pIwfp9LaeNhSu8O1YJfKModh5I7Nuzw/edit#gid=0) *(Constants used can also be referred in the previous***[***blog***](https://mes100.com/blog/graphs-and-equations-used-in-culvert-design-nomographs-part-full-flow-critical-depth-full-flow/)***post)* ![](https://i0.wp.com/mes100.com/wp-content/uploads/2019/05/Chart-18.A2.png?resize=1024%2C143&ssl=1) - [**Design Chart 18.A3 for**](https://docs.google.com/spreadsheets/d/1iJOwK4Ku6GeYtkFmFSgvHq_4X5wcknN9v5tWcEopDkM/edit#gid=0)**[Box](https://docs.google.com/spreadsheets/d/1iJOwK4Ku6GeYtkFmFSgvHq_4X5wcknN9v5tWcEopDkM/edit#gid=0)**[**Culvert**](https://docs.google.com/spreadsheets/d/1iJOwK4Ku6GeYtkFmFSgvHq_4X5wcknN9v5tWcEopDkM/edit#gid=0) *(Constants used can also be referred in the previous***[***blog***](https://mes100.com/blog/graphs-and-equations-used-in-culvert-design-nomographs-part-full-flow-critical-depth-full-flow/)***post)* ![](https://i0.wp.com/mes100.com/wp-content/uploads/2019/05/Chart-18.A3.png?resize=1024%2C133&ssl=1) # **Relative Discharge, Velocity and Hydraulic Radius in Part-full Pipe Culvert and Box Culvert Flow** Based on Design Chart 18.A5 and Design Chart 18.A6 Manning equation is used to relate with the part-full flow but with some modification of the roughness coefficient. | **y/D** | **n/nfull** | | --- | --- | | 0 < y/D <= 0.03 | 1 + (y/D)/(0.3) | | 0.03 < y/D <= 0.1 | 1.1 + (y/D - 0.03)(12/7) | | 0.1 < y/D <= 0.2 | 1.22 + (y/D - 0.1)(0.6) | | 0.2 < y/D <= 0.3 | 1.29 | | 0.3 < y/D <= 0.5 | 1.29 - (y/D - 0.3)(0.2) | | 0.5 < y/D <= 1 | 1.25 - (y/D - 0.5)(0.5) | You can also refer to the **[**blog**](https://mes100.com/blog/graphs-and-equations-used-in-culvert-design-nomographs-part-full-flow-critical-depth-full-flow/)** post regarding to this. Example of the calculation is provided for you to refer. - [**Design Chart 18.A5 for Pipe Culvert**](https://docs.google.com/spreadsheets/d/17Syk4cnKRCZDD3JxvMPUKLuVJwDyxbHUui34GIWDCRQ/edit#gid=114277767) ![](https://i0.wp.com/mes100.com/wp-content/uploads/2019/05/Chart-18.A5.png?resize=1024%2C153&ssl=1) - [**Design Chart 18.A6 for Box Culvert**](https://docs.google.com/spreadsheets/d/1JiBNyYi-iujmk_EeM9PpoR-JYKDjJzJhugXhbZCi0NI/edit#gid=0) ![](https://i0.wp.com/mes100.com/wp-content/uploads/2019/05/Chart-18.A6.png?resize=1024%2C367&ssl=1) # **Critical Depth in a Circular Pipe and Box Culvert** Based on Design Chart 18.A7 and Design Chart 18.A8 The equation of critical depth for pipe culvert is as follows; $$ \style{font-family:Arial}{{\mathrm h}_{\mathrm c}\;=\;\frac{\mathrm{CQ}^{1.5}}{\mathrm D^{0.25}}} $$ hc = Critical depth (m) C = Constant, 0.562 Q = Discharge (m3/s) D = Diameter (m) Equation used for box culvert is as follows; $$ \style{font-family:Arial}{{\mathrm h}_{\mathrm c}\;=\;\frac{\mathrm C.\mathrm Q^{\displaystyle\frac23}}{\mathrm B}} $$ hc = Critical depth (m) C = Constant, 0.319 Q = Discharge (m3/s) B = Width of culvert (m) Example of calculation to benchmark the values are as below; - [**Design Chart 18.A7 for Pipe Culvert**](https://docs.google.com/spreadsheets/d/1q6DAhaYCJ8pSmt4Xc1c8a2fDJLF_5j1GhaDL-SKfEqA/edit#gid=0) ![](https://i0.wp.com/mes100.com/wp-content/uploads/2019/05/Chart-18.A7.png?resize=546%2C330&ssl=1) - [**Design Chart 18.A8 for Box Culvert  **](https://docs.google.com/spreadsheets/d/1hmfoE2L1Wqxb-HXGKZMHl0BZmREfJ_Ue2FAEYkF4jsY/edit#gid=0) ![](https://i0.wp.com/mes100.com/wp-content/uploads/2019/05/Chart-18.A8.png?resize=551%2C330&ssl=1) # **Outlet Control Nomograph of Pipe Culvert and Box Culvert** Based on Design Chart 18.A9 and Design Chart 18.A10 Equation for outlet control nomographs also is derived as follows. Same equation can be used for both pipe and box culvert. The constants can be referred in the calculation or in previous **[**blog**](https://mes100.com/blog/graphs-and-equations-used-in-culvert-design-nomographs-part-full-flow-critical-depth-full-flow/)**. $$ \style{font-family:Arial}{\mathrm H\;=\;(1\;+\;\mathrm k<\mathrm{em}>\mathrm e\;+\;\frac{\mathrm k<\mathrm{/em}>\mathrm u.\mathrm n^2.\mathrm L}{\mathrm R^{1.33}})\;(\frac{\mathrm V^2\;}{2\mathrm g})} $$ H = Barrel loss (m) Ke = Constant Ku = Constant, 19.63 SI Unit n = Manning rougness coefficient L = Length of culvert barrel (m) R = Hydraulic radius of the full culvert barrel (m) V = Velocity in the barrel (m/s) g = Gravitational accelaration, 9.81 m/s/s SI unit Example of the calculations are as below; - [**Design Chart 18.A9 for Pipe Culvert**](https://docs.google.com/spreadsheets/d/1iUInwLCFcyOWee-MVbCLr0twi5Eif-ZsefrPN3ZtU8E/edit#gid=0) ![](https://i0.wp.com/mes100.com/wp-content/uploads/2019/05/Chart-18.A9.png?resize=1024%2C166&ssl=1) - [**Design Chart 18.A10 for Box Culvert**](https://docs.google.com/spreadsheets/d/1vgkngTIM7HnZIzOUaysffzDcZPiHd-Juyx0SBnlk1DE/edit#gid=0) ![](https://i0.wp.com/mes100.com/wp-content/uploads/2019/05/Chart-18.10.png?resize=1024%2C159&ssl=1) ### *Related* [![](https://i0.wp.com/mes100.com/wp-content/uploads/2024/10/me-small.jpg?resize=100%2C100&ssl=1)](https://mes100.com/author/admin/)[Ngu Soon Hui](https://mes100.com/author/admin/) I’m the [Benevolent Dictator for Life](https://en.wikipedia.org/wiki/Benevolent_dictator_for_life) for MiTS Software cum Editor of this website. Read more [here](/about-ngu-soon-hui/). You can also contact me at soonhui@mes100.com