U.S. Department of Transportation
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Federal Highway Administration Research and Technology
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Publication Number: FHWA-RD-06-138
Date: October 2006
Effects of Inlet Geometry on Hydraulic Performance of Box Culverts
APPENDIX D. EXAMPLE PROBLEM
The 25-year and 100-year floods at a 34.97-square-kilometer (km2) (13.5-square-mile (mi2)) design site in South Dakota have peak flows of 21.6 m3/s (773 ft3/s)(Q25) and 44.9 m3/s (1602 ft3/s) (Q100).
REQUIREMENT: Design and compare the headwater elevations for the Q25 and Q100 peak flows using a twin 2.7- by 2.4-m (9- by 8-ft) cast-in-place (field cast) culvert and a twin 2.7- by 2.4-m (9- by 8-ft) precast box culvert.
The low roadway grade has an elevation of 27.51 m (90.20 ft).
The downstream cross section ground point coordinates are given in table 20.
The tailwater rating information is given in table 21.
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.
Figure 138. Equation. Downstream channel velocity for Q25.
Figure 139. Equation. Downstream channel velocity for Q100.
Figure 140. Equation. Critical depth, below top corner fillets.
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.
Solving this equation would be a trial and error procedure, but it can be solved using the goal seek tool of Microsoft® Excel.
Figure 142. Equation. Normal culvert depth.
If the normal depth is below the top corner fillet, the flow area, A, and the hydraulic radius, Rh, 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, Rh, can be computed from the equations in figure 144.
Figure 144. Equations. Flow area and hydraulic radius, top fillets partially submerged.
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.
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 Ko = 1.0 (see HDS-5, p. 35).
The outlet is unsubmerged, and the footnote below table 6 of the research report warns that the unsubmerged Ko 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.
This step can be done fairly easily on a spreadsheet by increasing the depth by increments between do at the outlet and the full culvert depth, D. Compute the step length, ΔL, from the equation in figure 153. Figures 149-152 contain equations for preliminary calculations for the equation in figure 153.
Figure 149. Equations. For d less than (D-a).
Figure 150. Equations. For d less than D but greater than (D-a).
Figure 151. Equations. For d equal to D (the last iteration).
The friction slope, SF, for any step is computed from Manning's equation.
Figure 152. Equation. Friction slope.
The results of the equations in figures 149-152 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.
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).
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 (Ke) of culverts in example problem.
Figure 156. Equation. Entrance loss.
Figure 157. Equation. Headwater energy grade line.
The HWEGL’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, HWHGL in the headwater pool, use the relationships in figure 158.
Figure 158. Equation. Headwater hydraulic grade line.
The water surface elevation, HWelevation, 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.
For the four culvert configurations, results of the step backwater and entrance loss computations for the Q100 discharge only are illustrated in figures 159 through 162.
Figure 159. Diagram. FC-D-30 model, Q100 elevations.
Figure 160. Diagram. FC-D-0 model, Q100 elevations.
Figure 161. Diagram. PC-B model, 30.48-cm (12-inch) corner fillets, Q100 elevations.
Figure 162. Diagram. PC-B model, no corner fillets, Q100 elevations.
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.
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 Q25 or Q100 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.
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 HY-8 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 PC-B, 2.7- by 2.4-m (9- by 8-ft) culverts was computed with no fillets. The error derived by not accounting for the 30.48-cm (12-inch) fillets was more than 0.0305 m (0.10 ft) for the Q100 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 HY-8 could certainly account for the corner fillets, but significant additional computer coding would be required.
At the Q100 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 30-degree-flared wingwalls.