#### This article is concerned with the hydraulic aspects of designing a box culvert. It deals particularly with the aspects of sizing a box culvert to ensure that it’s able to deal with the anticipated peak flood discharge without any adverse effect throughout the service life of the culvert.

The efficient design of a box culvert must cover two principal aspects: the hydrological/hydraulic design and the structural design. Hydrological/hydraulic design is usually the foremost consideration. Why? Box culverts are not designed to be fancy; they serve as conduits for water flow beneath roads, railways, and other pathways. Thus, meticulous attention to hydrological consideration is paramount to ensure optimal performance. And, understanding the intricate interplay between precipitation patterns, runoff characteristics, and hydraulic behavior is essential for engineers tasked with designing box culverts that can withstand diverse hydrological conditions towards mitigating the risks of flooding.

Hydrological considerations play a fundamental role in determining the dimensions, alignment, and inlet/outlet configurations of box culverts. This requires an analysis of historical rainfall data, conducting hydrological modeling, and assessing local topography. Through this, engineers can gain valuable insights into the anticipated flow rates, velocities, and depths that a culvert must accommodate. Furthermore, accounting for factors such as watershed characteristics, soil permeability, and land use patterns enables engineers to develop tailored solutions that effectively manage water conveyance while minimizing the potential for erosion, sedimentation, and infrastructure damage.

This article is concerned with the hydraulic aspects of designing a box culvert. It deals particularly with the aspects of sizing a box culvert to ensure that it’s able to deal with the anticipated peak flood discharge without any adverse effect throughout the service life of the culvert.

**Sizing a Culvert**

The performance of a culvert design is the efficient transportation of flow from one side of a road to the other side. Thus, in order for a culvert to be sized, two very important parameters need to be determined. The first being the peak flow or discharge Q_{p }which is calculated on the basis of the characteristics of the catchment area – (the area surrounding the location where the culvert is to be found) and rainfall intensity. The second parameter is the discharge capacity of the culvert structure Q_{c }which is dependent on the material and geometrical properties of the culvert.

For a culvert to have been correctly sized, the peak discharge should never exceed the discharge capacity of the culvert throughout the service life of the culvert. This can be expressed as:

Q_p\le Q_c

**Peak Flood Discharge**

The peak flood discharge represents the maximum flow rate of water passing through a culvert during a flood event. Estimating the value of the peak flood discharge is crucial for designing culverts that can effectively manage and convey stormwater runoff while minimizing the risk of flooding. The determination of peak flood discharge involves complex hydraulic calculations considering factors such as the watershed characteristics, rainfall intensity, and topography.

Several methods have been devised to compute peak flood discharge, including the use of empirical formulas, charts, hydraulic modelling software etc. In this article, the rational method, put forward by Lloyd-Davis is presented. This is expressed as:

Q_p = 0.278CIA

Where:

- Q
_{p}is the peak flood discharge in m^{3}/s - C is the runoff coefficient expressed as percentage of imperviousness of the watershed surface. Its value depends on many variables such as the type and density of vegetation, type of surface and nature of soil, the slope of the terrain and so on.
- I = average intensity of rainfall measured in mm/hr for storm duration equal to the time of concentration to that is the time period required for the rain-water to reach the outlets from the most remote point of the catchment area.
- A = watershed area in square kilometers.

**Runoff Coefficient ‘C’**

Run-off coefficient represents the portion of rainfall that becomes surface runoff after accounting for various factors such as land use, soil type, slope, and vegetation cover. The run-off coefficient is a dimensionless parameter typically ranging between 0 and 1, where a value of 0 indicates complete infiltration of rainfall into the soil, and a value of 1 signifies that all rainfall becomes surface runoff.

The run-off coefficient essentially quantifies the hydrological response of a particular area to rainfall, providing valuable information for estimating stormwater runoff volumes and designing drainage systems. The run-off coefficient is there in the Loyd Davis formula to allow engineers more accurately predict the peak flood discharge.

Typical values of run-off coefficient based on type of terrain and slope can be obtained here. It is expected that many times a catchment area cannot be exactly defined by a singular but a combination of terrain and slope. In such instances, the run-off coefficients can be weighed to arrive at a more realistic value.

**Rainfall Intensity ‘I’**

Rainfall intensity refers to the rate at which precipitation falls onto a given area during a specific time period. It is a crucial parameter in estimating the peak flood discharge as it directly influences the amount and timing of surface runoff generated during a storm event. Rainfall intensity is typically expressed in units of depth per unit time, such as millimeters per hour or inches per hour. The intensity of rainfall can vary significantly depending on factors such as geographic location, weather patterns, and storm characteristics. High rainfall intensity can lead to rapid runoff and increased risk of flooding, making it essential to accurately estimate and account for in hydrological analyses and engineering designs.

Estimating rainfall intensity can be done using various methods and techniques, depending on the available data and the desired level of accuracy. One common approach is to use rainfall intensity-duration-frequency (IDF) curves, which provide relationships between the intensity, duration, and frequency of rainfall events based on historical precipitation data. These curves are typically developed using statistical analysis of rainfall records collected from weather stations over an extended period. Another method involves using radar or satellite-based rainfall estimates, which provide real-time or near-real-time information on rainfall intensity over a specific area. Additionally, rain gauge networks and weather radar systems can be utilized to monitor and measure rainfall intensity directly at specific locations.

Mathematically, rainfall intensity can be expressed as:

I=\frac{(A+B)log10n}{(t_c+a)^b}

Where: a; b; A; and B are station constants, while t_{c} is known as time of concentration.

The value of a, b, A and B can be easily gotten from meteorological agencies data or rain gauge network information. Time of concentration on the other hand is dependent on certain characteristics of the catchment. Mathemically, time of concentration can be calculated from:

t_c =0.078\frac{(L\times3.28)^{0.77}}{60S^{1/2}}

Where: t_{c} = time of concentration in (hours); L is the length of watershed area in m and S is the slope of watershed area in (m/m)

**Watershed Characteristics**

In sizing a box culvert, watershed characteristics must be considered. Recall that the catchments area A, and the length of watershed L and Slope of watershed are all necessary factors required to compute peak flood discharge.

Watershed characteristics encompass a range of natural features and attributes within the drainage area that influence runoff response to rainfall events. These characteristics include the size and shape of the watershed, land cover, soil type, topography, and drainage patterns. Each of these factors plays a significant role in determining how rainfall is distributed, infiltrated, and ultimately transformed into runoff within the watershed. For instance, watersheds with steep slopes and impermeable surfaces tend to generate higher runoff rates compared to flatter areas with more permeable soils and vegetation cover.

Three parameters are required to be determined here. First is the Catchment Area A, then the length of the watershed area, L and then the slope of the watershed area S.

Estimating this watershed characteristics typically involves field surveys, remote sensing techniques, and analysis of geographic information system (GIS) data. Alternatively, and for preliminary analysis, software such as Google Earth or Autodesk Infraworks can be utilized to determine these values. How to do this would be shown in the worked example.

**Culvert Capacity**

Having determined the peak flood design, the next step involves selecting a culvert section and then checking that the discharge capacity meets the requirement of the peak flood discharge. The Cason-Manning Equation comes in handy here.

The Cason Manning equation is a widely used method for calculating the discharge capacity of culverts. This empirical equation provides a straightforward approach to estimate the flow rate through culverts based on key geometric and hydraulic parameters. The equation is expressed as:

Q_c=\frac{1}{n}\times R^{2/3}\times S^{1/2}\times A

Where:

- Q is the discharge capacity (flow rate) of the culvert,
- n is the discharge coefficient (dimensionless),
- A is the cross-sectional area of flow (square feet or square meters),
- R is the hydraulic radius (feet or meters) which is the ratio of the area of the culvert section and the wetted perimeter of the culvert.
- S is the slope of the energy grade line (feet per foot or meters per meter). Which is usually a minimum of 1%.

The discharge coefficient (n) in the Cason Manning equation incorporates various factors affecting flow resistance within a culvert, such as shape, roughness, and entrance conditions. It is typically determined empirically based on experimental data or published tables for different culvert types and flow conditions (Typical values can be found here)

While the Cason Manning equation provides a simplified method for estimating culvert discharge capacity, it is important to note that its accuracy may vary depending on factors such as culvert shape, flow regime, and hydraulic conditions. Therefore, engineers often use it in conjunction with other hydraulic analysis methods and consider site-specific factors to ensure reliable and accurate culvert design and performance.

*Worked Example*

*Worked Example*

A Box culvert is required to be sited along a collector road in Abuja. The co-ordinate of the location is 326584.00E and 995589N. Using google earth pro the catchment area has been estimated as 19.94km^{2}, the maximum length of watershed area is 7.76km and the maximum and minimum elevation across the watershed area is 458m and 420m respectively (See Figure). Assuming the average precipitation during a storm, for the location, taken over 25years period is 1525mm. Determine the adequacy of a Double Cell 3m x 3m Box culvert for the location.

N: B The following are constants for the rain gauge station closest to the proposed culvert location and provided by the meteorological service department of Nigeria NIMET:

*a =0.500; b=1.032; A=2.950; B=1.910*

#### Peak Flood Discharge

According to Llyod-Davis equation, the peak flood discharge can be estimated as follows:

Q_p=0.278CIA

Three parameters are required – runoff coefficient, rainfall intensity and catchment area. We have the catchment area A = 19.94km2, but we must estimate run-off coefficient and rainfall intensity.

##### Run-off Coefficient

By inspection, the catchment area consists of a combination of terrain, hence the run-off coefficient can be weighted. The run-off coefficient will be calculated considering a rolling area covered for the 60% by cultivated land day & loam land (run-off coefficient = 0.55) and for the remaining 40% by unimproved lands (run -off coefficient = 0.20). Thus, the average run-off coefficient is.:

0.60 \times 0.55 + 0.40 \times 0.20 = 0.41

**Rainfall Intensity **

I=\frac{(A+B)log10n}{(t_c+a)^b}

To determine the rainfall intensity, we must determine the time of concentration, which is given as:

t_c =0.078\frac{(L\times3.28)^{0.77}}{60S^{1/2}}

L=7.76km

S=\frac{\Delta H}{L} =\frac{458-420}{7760}=4.9\%

t_c=0.078\frac{(7760\times3.28)^{0.77}}{(60\times0.0049)^{1/2}} =4.56hrs

Therefore, rainfall intensity is:

I_{25}=\frac{(2.95+1.91)log10^{25}}{(4.56+0.50)^{1.032}}=63.27mm/hr

Having determined the parameters required to estimate the peak flood discharge, we can then apply the Loyd-Davis equation as follows:

Q_p=0.278CIA

=0.278\times0.41\times63.27\times19.94= 143.8m^3/s

#### Culvert Capacity

Having determined the peak flood discharge, the next step is to determine the capacity of the culvert to discharge this volume of water, by applying the Cason-Manning equation:

Q_c=\frac{1}{n}\times R^{2/3}\times S^{1/2}\times A

The box culvert consists of two cells of 3m x 3m, therefore, Area of section:

A= 2\times (3.0\times3.0)=18m^2

The Hydraulic radius is given as ratio of the area A to the wetted perimeter w:

R=\frac{A}{w}

The wetted perimeter is basically the area that is in contact with water when the box culvert is full. This is given as:

w=2\times(2\times(3+3)= 24m

R=\frac{18}{24} =0.75m

Adopting a minimum slope of 1% and assuming the box culvert is made out of concrete so that the manning coefficient is =0.012. the discharge capacity of the culvert is:

Q_c=\frac{1}{0.012}\times0.75^{2/3}\times 0.001^{1/2}\times 18 \\= 184.00m^3

(Q_p=143.8m^2) \le (Q_c=184.00m^2)

Therefore, since the discharge capacity of the proposed culvert exceeds the peak flood discharge value, we can conclude that a **(3m x 3m) Double cell box culvert** would be satisfactory for the proposed location.

It is, however, important to state here that this analysis has been carried out on the assumption that there are no upstream structures. If there was an upstream structure, and it can be assumed that the upstream structure was adequately sized, the discharge capacity of that upstream structure should be added to the peak flood discharge, in checking the capacity of the proposed culvert. In that case the catchment area contributing flow to the upstream structure would be ignored.

Also See: How to Apply Loads on Box Culverts| Eurocode

### Sources & Citations

- Kilgore, R. T., Morris, J. L., Schall, J. D., Thompson, P. L. and Zerges, S. M. (2012). “Hydraulic Design of Highway Culverts Third Edition”. Federal Highway Administration (FHWA), Washington, D.C. PP 326.