Effects of Inlet Geometry on Hydraulic Performance of Box Culverts
APPENDIX D. EXAMPLE PROBLEM
The 25year and 100year floods at a 34.97squarekilometer (km^{2}) (13.5squaremile (mi^{2})) design site in South Dakota have peak flows of 21.6 m^{3}/s (773 ft^{3}/s)(Q_{25}) and 44.9 m^{3}/s (1602 ft^{3}/s) (Q_{100}).
REQUIREMENT: Design and compare the headwater elevations for the Q_{25} and Q_{100} peak flows using a twin 2.7 by 2.4m (9 by 8ft) castinplace (field cast) culvert and a twin 2.7 by 2.4m (9 by 8ft) precast box culvert.
The low roadway grade has an elevation of 27.51 m (90.20 ft).
Given:
Elevation of inlet invert:  24.04 m (78.81 ft) 
Elevation of outlet invert:  24.03 m (78.79 ft) 
Culvert length:  25.62 m (84 ft) 
Stream bed slope:  0.02 percent 
The downstream cross section ground point coordinates are given in table 20.
Table 20. Example problem, downstream cross section ground point coordinates.
X (ft)  YElevation (ft)  Comment 
64  86.0  
130  84.0  
152  83.5  
197  83.0  
245  82.5  
277  82.0  
293  81.0  Edge of channel 
300  78.7  
305  81.0  
329  82.0  
406  82.5  Edge of channel 
470  83.0  
500  86.0  
The tailwater rating information is given in table 21.
Table 21. Example problem, tailwater rating information.
Flow (ft^{3}/s)  Tailwater elevation (ft) 
68.7  82.77 
222.0  83.49 
375.4  83.93 
528.7  84.30 
682.0  84.61 
773.0  84.78 
988.7  85.15 
1142.0  85.38 
1295.3  85.61 
1448.7  85.82 
1602.0  86.00 
1 ft = 0.305 m; 1 ft^{3}/s = 0.028 m^{3}/s 
 
STEP 1:  Plot the downstream discharge rating curve and flow area curves based on ground point coordinates. 
Figure 133. Graph. Discharge, tailwater variation.
Figure 134. Graph. Downstream cross section.
Figure 135. Graph. Cross section area versus tailwater elevation.
Based on a regression of the area versus the tailwater elevation curve in figure 135, the downstream flow area for tailwater elevation is given by the equation in figure 136.
Figure 136. Equation. Downstream flow area for tailwater elevation.
Assuming the upstream section is a vertical shift of the downstream section according to the 0.02 percent channel slope, the flow area under the headwater elevation can be computed by the equation in figure 137.
Figure 137. Equation. Flow area under headwater elevation.
 
STEP 2:  Compute downstream channel velocity, V_{TW}. 
Figure 138. Equation. Downstream channel velocity for Q_{25}.
Figure 139. Equation. Downstream channel velocity for Q_{100}.
 
STEP 3:  Compute critical depth using the equation in figure 140. 
Figure 140. Equation. Critical depth, below top corner fillets.
Where:  
 
B  is total culvert width; NB times span of each barrel. 
d_{c}  is flow depth measured from the invert. 
a  is the corner fillet height; 0.153 m (0.5 ft) for the FC culvert, and 0.305 m (1 ft) for the PC culvert. 
NB  is the number of barrels. 
The equation in figure 140 applies if the critical depth is below the top corner fillets. If the critical depth does partially submerge the top fillets, the relationship becomes the equation in figure 141.
Figure 141. Equation. Critical depth, partially submerged top corner fillets.
Where:  
a_{t}  is the submergence of top corner fillets; d_{c}(Da). 
Solving this equation would be a trial and error procedure, but it can be solved using the goal seek tool of Microsoft® Excel.
Table 22. Example problem, step 3 solutions.
Culvert (ft by ft)  a (inches)  d_{c} for Q_{25} (ft)  d_{c} for Q_{100} (ft)  Critical depth elevation at outlet for Q_{25} (ft)  Critical depth elevation at outlet for Q_{100}(ft) 
FCD30 9 by 8  6  3.88  6.29  82.67  85.09 
FCD0 9 by 8  6  3.88  6.29  82.67  85.09 
PCB 9 by 8  12  3.96  6.37  82.75  85.16 
PCB 9 by 8  0  3.86  6.26  82.65  85.05 
1 inch = 2.54 cm; 1 ft = 0.305 m 
 
STEP 4:  Determine normal depths in the culvert from Manning's equation (figure 142). 
Figure 142. Equation. Normal culvert depth.
If the normal depth is below the top corner fillet, the flow area, A, and the hydraulic radius, R_{h},
can be computed from the equations in figure 143.
Figure 143. Equations. Flow area and hydraulic radius, depth below top corner fillet.
If the normal depth partially submerges the top corner fillets, the flow area, A, and the hydraulic radius, R_{h}, can be computed from the equations in figure 144.
Figure 144. Equations. Flow area and hydraulic radius, top fillets partially submerged.
Where:  
 
a_{t}  is the submergence of top corner fillets; d_{c}(Da). 
If the normal depth exceeds the rise, D, of the culvert, set at equal to a and compute the normal depth that would occur if the culvert did not have a crown.
The normal depth can be determined by trial and error or by using the goal seek tool from Excel.
Table 23. Example problem, step 4 solutions.
Culvert (ft by ft)  a (inches)  d_{n} for Q_{25} (ft)  d_{n} for Q_{100} (ft) 
FCD30 9 by 8  6  10.49  19.61 
FCD0 9 by 8  6  10.49  19.61 
PCB 9 by 8  12  10.69  19.82 
PCB 9 by 8  0  10.44  19.57 
1 inch = 2.54 cm; 1 ft = 0.305 m 
 
STEP 5:  Determine initial depth do at barrel exit to start backwater calculation. 
The normal depths are greater than critical depths; therefore, the culverts will be outlet control whether or not the barrels flow full at the inlet. The tailwater elevations are greater than the critical depth elevations and are below the crown elevations at the outlet; therefore, the depth at the culvert outlet will be between the critical depth and the culvert crown and can be computed from the equation in figure 145.
Figure 145. Equation. Initial depth.
Assume K_{o} = 1.0 (see HDS5, p. 35).
The outlet is unsubmerged, and the footnote below table 6 of the research report warns that the unsubmerged K_{o} values are unreliable. Nevertheless, the unsubmerged value for a twin box culvert happens to be the traditional value that is recommended for the outlet loss. The exit coefficients derived for this study neglect the tailwater velocity head.
Figure 146. Equation. For Ko equals 1.0.
Figure 147. Diagram. Definition sketch for exit loss.
Figure 148. Equation. Initial depth, ignoring tailwater velocity head.
Table 24. Example problem, step 5 solutions.
Culvert (ft by ft)  TW HGL elevation for Q_{25} (ft)  d_{o} for Q_{25} (ft)  EGL elevation at culvert outlet for Q_{25} (ft)  TW HGL elevation for Q_{100} (ft)  d_{o} for Q_{100} (ft)  EGL elevation at culvert outlet for Q_{100} (ft) 
FCD30 9 by 8  84.78  5.99  85.58  86.00  7.21  88.38 
PCB 9 by 8  84.78  5.99  85.61  86.00  7.21  88.46 
 
STEP 6:  Use standard step backwater calculations to determine the EGL in the culvert for free surface flow. 
This step can be done fairly easily on a spreadsheet by increasing the depth by increments between d_{o} at the outlet and the full culvert depth, D. Compute the step length, ΔL, from the equation in figure 153. Figures 149152 contain equations for preliminary calculations for the equation in figure 153.
Figure 149. Equations. For d less than (Da).
Figure 150. Equations. For d less than D but greater than (Da).
Figure 151. Equations. For d equal to D (the last iteration).
Where:  
 
subscript "i"  is a line in the stepbackward computation. 
a_{t}  is the partially submerged top fillet. 
d  is the flow depth in the barrel. 
D  is the rise of the culvert. 
The friction slope, S_{F}, for any step is computed from Manning's equation.
Figure 152. Equation. Friction slope.
Where:  
 
A_{m}  is (A_{i} + A_{i1})/2. 
R_{hm}  is (R_{hi} + R_{hi1})/2. 
n  is the Manning roughness coefficient: 0.008 m^{1/3} (0.012 ft^{1/3}). 
Q  is discharge. 
The results of the equations in figures 149152 enable the calculation of the step length, ΔL. The step length may also be viewed as calculated from the energy balance.
Figure 153. Equation. Step length.
Where:  
 
V  is flow velocity in barrel. 
S_{b}  is barrel slope. 
After each step length calculation, the ΔL's are summed to give L, which is compared with the length of the culvert (25.62 m (84 ft)) to determine when computations are complete. Since the corner fillets affect the backwater computations, the calculations should be done in two stages.
If the culvert fills to the crown before the computations reach the entrance, the EGL at the entrance, or upstream end, is given by the equation in figure 154.
Figure 154. Equation. EGL at upstream culvert end (the entrance).
Where:  
 
V_{FULL}  is Q/A_{FULL}. 
A_{FULL}  is NB((span)Da^{2}). 
R_{hFULL}  is A_{FULL}/(NB(span + 2D2.343a)). 
L_{FULL}  is the length of a culvert flowing full. 
 
STEP 7:  Determine entrance loss. 
HGL is greater then critical depth throughout the barrel, indicating outlet control. Use table 11 (in chapter 7) to find entrance loss coefficients. The sketches in figure 155 are from figure 93 in chapter 7.
Figure 155. Sketches. Entrance loss coefficients (K_{e}) of culverts in example problem.
Figure 156. Equation. Entrance loss.
Where:  
 
V_{US}  is velocity at the upstream end of the culvert from the step backwater or the full flow computations. 
 
STEP 8:  Compute headwater elevation. 
Figure 157. Equation. Headwater energy grade line.
Where:  
 
EGL elevation_{US}  is the EGL elevation at the upstream end of the culvert from the backwater calculations 
The HW_{EGL}’s are energy grade line elevations and include the velocity head, which is usually negligible in the headwater pool. To get the actual water surface elevations, HW_{HGL} in the headwater pool, use the relationships in figure 158.
Figure 158. Equation. Headwater hydraulic grade line.
Where:  
 
A  is computed from the equation in figure 137. 
The water surface elevation, HW_{elevation}, in the headwater pool can be determined by trial and error or by using the goal seek tool of Excel.
Tables 25 and 26 summarize the results of the step backwater computations, the headwater EGL computations, and the headwater HGL computations.
Table 25. Example problem, step backwater and entrance loss results for Q_{25}.
Culvert  Span,Rise  Fillet Size (inches)  HGL Elevation_{US} (ft)  EGL Elevation_{US} (ft)  K_{e}  HW_{EGL}(ft)  Area in HW Pool (ft^{2})  Velocity in HW Pool (ft/s)  Water Surface Elevation HW_{elevation} (ft) 
FCD30  9, 8  6  84.877  85.662  0.32  85.913  1265  0.65  85.907 
FCD0  9, 8  6  84.877  85.662  0.52  86.071  1335  0.58  86.065 
PCB  9, 8  12  84.879  85.686  0.54  86.121  1358  0.57  86.116 
PCB  9, 8  0  84.879  85.656  0.54  86.076  1228  0.63  85.823 
1 inch = 2.54 cm; 1 ft = 0.305 m 
Table 26. Example problem, step backwater and entrance loss results for Q_{100}.
Culvert  Span,Rise  Fillet Size (inches)  HGL Elevation_{US} (ft)  EGL Elevation_{US} (ft)  K_{e}  HW_{EGL}(ft)  Area in HW Pool (ft^{2})  Velocity in HW Pool (ft/s)  Water Surface Elevation HW_{elevation} (ft) 
FCD30  9, 8  6  86.458  88.578  0.32  89.256  3028  0.53  89.251 
FCD0  9, 8  6  86.458  88.578  0.52  89.680  3294  0.48  89.676 
PCB  9, 8  12  86.471  88.657  0.54  89.837  3394  0.47  89.834 
PCB  9, 8  0  86.463  88.563  0.54  89.697  3305  0.48  89.693 
1 inch = 2.54 cm; 1 ft = 0.305 m 
For the four culvert configurations, results of the step backwater and entrance loss computations for the Q_{100} discharge only are illustrated in figures 159 through 162.
Figure 159. Diagram. FCD30 model, Q_{100} elevations.
Figure 160. Diagram. FCD0 model, Q_{100} elevations.
Figure 161. Diagram. PCB model, 30.48cm (12inch) corner fillets, Q_{100} elevations.
Figure 162. Diagram. PCB model, no corner fillets, Q_{100} elevations.
SUMMARY
This example is included to illustrate how to apply the entrance loss coefficients from the laboratory results. SDDOT provided site data and culvert options. The example was divided into eight basic steps for spreadsheet computations because no design program accounted for the corner fillets, which were a consideration in the laboratory study.
The first step was to plot the channel cross section and derive expressions for channel area versus water surface elevations. The tailwater channel velocities and EGL elevations were then computed from the downstream rating data that were provided. The critical depth and normal depth in the culvert were computed to determine if inlet control was a possibility. The normal depth computation is a tedious trial and error process, especially when corner fillets are included in the computation, but the goal seek tool from Excel makes the task easier. Because normal depths were greater than critical depths, inlet control was eliminated as a possibility. The brink depth at the culvert outlet was computed from the equations in figure 163.
Figure 163. Equations. Brink depth at culvert outlet.
Where:  
 
K_{o}  is the outlet loss coefficient, assumed to be 1.0. 
Since the brink depth was below the crown of the culvert, a spreadsheet was developed for step backwater computations through the culvert by increasing the depth by increments and computing the corresponding step length. Spreadsheets were analyzed to determine where either the cumulative step lengths equaled the culvert length or the culvert flowed full. None of the culverts flowed full for either Q_{25} or Q_{100} before the cumulative step lengths equaled the culvert length of 25.62 m (84 ft); thus free surface flow occurred in each case. From the step backwater computations, the velocity and energy grade line elevation at the upstream end were used to compute the headwater energy grade line elevation from the equation in figure 164.
Figure 164. Equation. Headwater EGL.
Where:  
 
EGL_{US} and V_{US}  are energy grade line elevation and velocity at the upstream end of the culvert from the step backwater computations. 
K_{e}  is the entrance loss coefficient from the laboratory results: 0.32 for the field cast inlet with 30degreeflared wingwalls; 0.52 for the field cast inlet with 0degreeflared wingwalls; and 0.54 for the precast inlet. 
Finally, the hydraulic grade line elevation (water surface elevation) was computed by subtracting the velocity head in the headwater pool from the energy grade line elevation. Because of the irregular channel geometry, this was also a trial and error computation and was accomplished with the Excel goal seek tool. There was very little difference between the energy grade line and water surface elevations in the headwater pool.
The net area (with the areas of fillets removed) was used for the step backwater computations and for velocity computations. The research showed that the coefficients were not affected by the fillet sizes tested as long as the net area was used in the computations. Figure 165 is a sketch showing the procedure for calculating the net area. The current version of the FHWA HY8 program does not account for corner fillets.
Figure 165. Diagram. Net area used for backwater computations.
To show the sensitivity of including or not including the corner fillets, the headwater elevation for the PCB, 2.7 by 2.4m (9 by 8ft) culverts was computed with no fillets. The error derived by not accounting for the 30.48cm (12inch) fillets was more than 0.0305 m (0.10 ft) for the Q_{100 }discharge. These errors would increase as the size of the culvert decreases and would decrease as the size of the culvert increases. A design program such as HY8 could certainly account for the corner fillets, but significant additional computer coding would be required.
At the Q_{100} discharge, the HW elevations for the precast culverts were approximately 0.183 m (0.6 ft) higher than the elevations for the field cast culverts with the 30degreeflared wingwalls.
Previous  Contents  Next
