6.5 Defining Bridge Cross Sections and Coefficients
To properly model the effects of a bridge, the full reach of river (both upstream and downstream) affected by the structure must be included in the analysis. The presence of the bridge causes flow to start contracting toward the bridge opening well upstream of the structure, and the expansion of flow out of the bridge continues even further downstream. Bridge modeling requires determining the proper cross-section locations for the start of contraction (section 4) and end of expansion (section 1), assigning appropriate expansion and contraction coefficients through this reach, and developing ineffective flow areas around the bridge.
Cross-Section Location Techniques
In large floods, the entire floodplain on either side of the bridge is often under water. The width of the bridge opening is almost always significantly smaller than the width of the valley, thus flow first contracts to pass through the bridge opening and then expands downstream. There are energy losses associated with this contraction and expansion, and these losses are usually greater than expansion and contraction losses between normal valley cross sections.
Locating the cross sections at the beginning of the contraction and the end of the expansion is based on the modeler's judgment, supplemented by limited published guidance. Historically, a common technique for locating the two cross sections has been the rule of thumb. This technique applies a contraction ratio (CR) into the bridge of 1:1 and an expansion ratio (ER) out of the bridge of 1:4. This relationship is interpreted as follows: For every 4 feet or meters downstream of the bridge, the effective flow width expands out one foot or meter on each side of the bridge. Figure 6.13 illustrates this concept. USACE has used this general rule since the early 1960s, although improved techniques are now available to better estimate these cross-section locations.
Figure 6.13 Traditional contraction and expansion ratios through a bridge opening.
The U.S. Geological Survey (USGS) uses different techniques to locate the bounding cross sections at a bridge crossing. The USGS locates the beginning of the contraction (cross section 4) at one bridge opening width upstream. The location of the end of the expansion (cross section 1) can vary, but is most often one bridge opening width downstream of the bridge face. For example, if the bridge opening width is 400 ft, the beginning of the contraction would be located 400 ft upstream and the end of expansion would be 400 ft downstream. Figure 6.14 shows the sections located using the USGS procedures. These sections may be at very different locations, compared to the locations determined using USACE's contraction and expansion ratios. There is no reason to select one rule of thumb over the other; it is the modeler's responsibility to appropriately select cross-section locations based on site constraints and knowledge of the flow patterns.
Figure 6.14 Location of expansion and contraction sections, USGS techniques.
Contraction Lengths and Ratios.
To provide a more accurate and defensible method of selecting locations for the expansion and contraction sections, the Army Corps of Engineers' Hydrologic Engineering Center published guidance (USACE, 1995) that gives a more scientific basis for selecting both cross-section locations and the corresponding loss coefficients. Actual data for five flood events at three bridges in Alabama and Mississippi were used to calibrate two-dimensional unsteady flow models. These models were used to study where the effective start of contraction and end of expansion occurred for these bridge locations and flood events. The models were used to develop generalized two-dimensional models for small, medium, and wide bridge openings, compared to the floodplain width. The studies showed that the start of contraction and end of expansion at bridges were not as far from the bridge as had been assumed with the USACE rule of thumb. Studies of narrow, medium, and wide bridge openings, comparing bridge width to active floodplain flow width, showed that the contraction ratio can be as low as 1:0.5 upstream and that the downstream expansion can end at a ratio of from 1:1 to 1:3. Table 6.3 provides a summary of the results for contraction ratios for various channel slopes and Manning's n ratios.
|
Channel Slope (s0), ft/mi
|
nob/nc = 1
|
nob/nc = 2
|
nob/nc = 4
|
|---|---|---|---|
|
1
|
1.0-2.3
|
0.8-1.7
|
0.7-1.3
|
|
5
|
1.0-1.9
|
0.8-1.5
|
0.7-1.2
|
|
10
|
1.0-1.9
|
0.8-1.4
|
0.7-1.2
|
The variables nob and nc represent the Manning n values for the overbank and channel, respectively, and s0 is the channel invert slope. Natural channel and floodplain situations with equal channel and overbank roughness (described by the values in the second column of Table 6.3), are seldom encountered. The overbank is normally considerably rougher than the channel; therefore, the ratios in columns 3 and 4 are more typical. As illustrated in the table, the contraction ratio (CR) rule of thumb of 1:1 is generally adequate for situations in which the roughness ratios are 2-4.
The studies also demonstrated that increasing contraction ratios are associated with increasing discharges. Additionally, the work developed equations for estimating both the CR and the length of contraction (Lc). Somewhat surprisingly, the contraction length correlated better with the actual results when downstream Froude Numbers at sections 1 and 2 were used rather than at sections 3 and 4. The best prediction equation for the length of contraction was determined to be the following, but should be used only when the CR is within the ranges presented in Table 6.3:
(6.8)
It was found that Equation 6.8 has an adjusted determination coefficient (R2) of 0.87 and a standard error of estimate (Se) of 31 feet. The adjusted determination coefficient reflects the amount of variance in the actual contraction length captured by Equation 6.8 (87 percent). The standard error of estimate means that, on average, use of Equation 6.8 for the computation of contraction length will result in one-third of the estimates being more or less than 31 feet from the actual value. Both these values are indicative of a statistically sound equation.
In SI units, the prediction equation for Lc is
(6.9)
The units of Equation 6.9 are m and m3/s. The adjusted determination coefficient is 0.87 and the standard error is 9.6 m. The modeler can use either of these equations with confidence when the stream being modeled has variables that fit the guidelines for which Equation 6.8 and Equation 6.9 were derived. The ranges of the variables are s0 = 1-10 ft/mi (0.2-2 m/km), floodplain width of about 1000 feet (300 m), bridge widths of 100-500 feet (30-150 m), and discharges of 5000-30,000 ft3/s (140-850 m3/s).
Example 6.2 Computing the length of contraction with Equation 6.8.
A bridge crossing a stream is to be modeled with HEC-RAS. The reach has the following properties upstream of the bridge:
Solution
The discharge is only slightly outside the range of flow data for which the equation was developed and is judged acceptable for this example. An initial value of the contraction ratio (CR) is needed to develop a location of cross section 4 so that the HEC-RAS model can be coded and operated. With the model output, the contraction ratio may then be refined using Equation 6.8.
First, estimate the parameters required to determine CR from Table 6.3:
The stream slope is given, therefore the ratio of the n values is needed (nob/nch = 0.09/0.045 = 2). From Table 6.3, CR (for the higher discharges) is interpolated as 1.4-1.5. Select an initial value of 1.5. If the bridge is symmetrical, both embankments are about 400 ft long [LOBS = (1200 - 400)/2 = 400 ft]. Therefore, the initial location for the cross section at the beginning of the contraction (section 4) is 1.5 × 400 = 600 ft upstream of the bridge face. This value is used for the initial HEC-RAS model and the program is operated for the selected discharge. From the model output, the variables required to compute the contraction length with Equation 6.8 are selected. This equation is appropriate because the floodplain width, bridge width, and discharge are close to the values used to derive Equation 6.8. From the HEC-RAS output for sections 1 and 2 (downstream of the bridge), the following values are found:
Inserting these values into Equation 6.8 gives
.
The computed contraction length is much smaller than the initial estimate of 600 ft. Therefore, the HEC-RAS data set is modified, with section 4 relocated to 424 ft upstream of the bridge and any geometry changes caused by the relocation made to the section. The program is run again and the new values for channel Froude numbers at sections 1 and 2 and the discharge in the overbank area at section 1 were unchanged, as would be expected for changes made upstream for this subcritical flow computation. The computed value of Lc is thus unchanged. No further modification to the location of section 4 is necessary.
For problems in which the variables are significantly out of these ranges, applicable to Equations 6.8 and 6.9, the following equation can be applied for the English system only. No similar equation was developed for SI units, although the modeler can convert the SI flowrate values to ft3/s to use this equation:
(6.10)
All variables are as defined in the previous equation, with CR being the contraction ratio. This equation has an adjusted determination coefficient of 0.65 and a standard error estimate of 0.19; therefore, it contains considerably more error (uncertainty) in the estimate of CR than does Equation 6.8 for contraction length.
Once a CR is calculated, the contraction length (Lc) is determined by multiplying the CR by the obstruction length. Assuming that the bridge opening is in the center of the cross section, the obstruction length equals (floodplain width - bridge width)/2.
When developing the location of the start of the contraction, tentative locations should be selected for an initial model run. The HEC-RAS model can then be executed and initial values for use in Equation 6.8 or Equation 6.10 can be obtained from the HEC-RAS output. All the variables in the two equations can be found in the detailed cross-section output from the program. The start of contraction can be found directly with Equation 6.8, or from the CR obtained from Equation 6.10 used in conjunction with a topographic map of the bridge reach. Depending on the range of geometry and discharge values obtained for the bridge being modeled, the engineer should use the appropriate equation to develop the final location for the section defining the start of contraction. Typically, only an initial estimate and one modification, to reflect the equation results, are required to properly locate the start of contraction. Values derived with either equation should be checked against the range of ratios shown in Table 6.3. Computed CR values less than 0.3 or more than 2.5 should be adjusted to values that are more reasonable.
Example 6.3 Computing the contraction ratio when Equation 6.8 is not applicable.
A bridge crossing a stream is to be modeled with HEC-RAS. The bridge reach has the following properties:
Solution
As in Example 6.1, an initial value of the contraction ratio (CR) is needed so that the HEC-RAS model can be coded and operated. However, the floodplain width and bridge width are not within or near the range of widths for which Equation 6.8 was developed; nor is the stream slope. Consequently, Equation 6.8 cannot be used to compute contraction length directly for this bridge. For streams with values outside of those for which Equation 6.8 was derived, Equation 6.10 is used to determine CR, using model output after an initial HEC-RAS run. The contraction length is determined by multiplying the CR and the obstruction length.
First, make an initial estimate of the parameters required to determine CR from Table 6.3. For a stream slope greater than 10 ft/mi and a ratio of overbank to channel n of 1.6, a CR is interpolated from Table 6.3. For higher discharges, a CR of 1.6 is appropriate for a stream slope of 10 ft/mi. Because the study stream is more than twice as steep, a lower CR can be used, since CR appears to decrease as slope increases. However, to be conservative, use CR = 1.6. If the bridge opening is in the center of the cross section, the obstruction length is (220 - 60)/2 = 80 ft. The first estimate of the contraction distance is 1.6 × 80 = 128 ft, which is then coded to the HEC-RAS model as the initial estimate of the contraction length. HEC-RAS is then operated and the detailed output at sections 1 and 2, just downstream of the bridge, and section 4 are reviewed to obtain the additional parameters for Equation 6.10. From the HEC-RAS output,
Applying Equation 6.10 gives the contraction ratio as
.
The revised contraction length is 0.74 × 80 = 59 ft, or less than half the initial estimate. The distance from the upstream bridge face to the beginning of contraction (section 4) is adjusted to the new value in the HEC-RAS model, the cross-section geometry is modified, if necessary, and the program rerun. Any significant revision in the overbank discharge at section 4 with the revised distance should be reapplied to Equation 6.10 to check for any additional adjustments in CR and contraction length. For this problem, no additional length adjustments may be required. The CR value computed with Equation 6.10 should be reasonable, with a range of allowable CR between 0.3 and 2.5.
Note that Equation 6.8 and Equation 6.10 both contain discharge terms. This indicates that the contraction distance or ratio will be different for every discharge analyzed. Many hydraulic analyses, such as flood insurance studies, must evaluate several water surface profiles, which can result in different geometric models for each discharge studied. Needless to say, managing this many data sets can be rather cumbersome. Therefore, for multiple profiles, a practical solution is to compute a contraction length, CR, based on an "average" flood discharge, or based on the largest discharge that does not greatly overtop the bridge or the embankments. In most cases, this average length is then used for analysis of all flood events. The modeler may wish to validate the use of the average length by performing sensitivity tests on the effect of the water surface elevation through the bridge by using varying contraction lengths with discharge. If the elevation difference between using a specific or an average value of contraction length is significant when compared to using separate lengths for each discharge, separate geometric models for each discharge may be necessary. In the experience of the author, the location of the start of the contraction section can vary by 100-200 ft (30-60 m) or more, and not result in a significant change in the computed water surface elevation through the bridge reach. The information in this paragraph is also applicable for computing an expansion length or ER, which is presented in more detail in the following subsection.
Expansion Lengths and Ratios.
With the same USACE data as in the preceding section, expansion ratios (ER) were found to be a function of channel roughness and slope, with bridge opening and floodplain widths also proving to be important factors. Expansion ratios were all considerably less than those given by the 1:4 rule of thumb, with the majority computed as less than 1:3. Table 6.4 shows the ranges of ER found for a variety of bridge conditions. In this table, b is the width of the bridge opening and B is the total floodplain flow width, as illustrated in Figure 6.1.
Similar to contraction ratios, the greater the discharge, the higher the expansion ratio. The mean of all the values is approximately 1.5, indicating that the traditional value of 1:4 is very conservative. Overbank roughness is normally considerably greater than channel roughness, so the last two columns of Table 6.4 represent more typical expansion ratios. The expansion ratios are generally no more than 1:2, much lower than past rule-of-thumb estimates.
Equations for expansion length and for ER were also developed. For English units, the best of the two prediction equations for expansion length is
| where |
Le |
= |
the length of the expansion reach (ft) |
| Lobs |
= |
the average length of the obstruction caused by the two bridge approaches (ft) |
(6.11)
The adjusted determination coefficient is 0.84 and the standard error of estimate is 96 feet, indicating that the equation is statistically sound.
For SI units, Equation 6.11 takes the form
(6.12)
The lengths are in meters, and the discharge is in m3/s. The adjusted determination coefficient is 0.84 and the standard error is 29.3 m.
Equation 6.11 and Equation 6.12 are applicable at bridge sites having parameters in the same range as was used to derive the equations. Parameter limits were discussed in the section above for contraction lengths and ratios.
Example 6.4 Computing the expansion length with Equation 6.11.
Compute the expansion reach for the bridge site of Example 6.2. An initial value of the expansion ratio (ER) is needed so that the HEC-RAS model may be coded and operated. With the model output, the expansion ratio may then be refined using Equation 6.11.
Solution
First, estimate the parameters required to determine the ER from Table 6.4.
The stream slope is 8 ft/mi and the ratio of the n values was found to be 2 in Example 6.2. The bridge opening ratio is computed as 400/1200 or 0.33. From Table 6.4, the ER ranges from approximately 1.2-2.0 for these values. Because the higher discharges reflect higher values of ER, an initial value for ER is selected as 2. The average abutment length (obstruction length) was found to be 400 ft in Example 6.2. Therefore, the initial location for the cross section at the beginning of the expansion (section 4) is 2 × 400 = 800 ft downstream of the bridge face. This value is used in the initial HEC-RAS model and from the model output, the variables required to compute the expansion length using Equation 6.11 are selected. This equation can be used because the floodplain width, bridge width, and discharge are within or close to the range of values used to derive Equation 6.11. The Froude numbers for the channel at sections 1 and 2 were found in Example 6.1, allowing Equation 6.11 to be applied to give
This computed value is approximately 10 percent greater than the initial estimate of Le. Therefore, the distances in the HEC-RAS model between the downstream bridge face and the end of expansion should be increased to 873 ft. Any cross-section geometry changes caused by the relocated section are made and the program is rerun. If there are significant changes in channel Froude numbers at sections 1 and 2, these values should be used in Equation 6.11 to recompute Le and determine if there are significant alterations in calculated water surface elevations through the bridge with the adjusted lengths. Normally, the initial revision for expansion length is all that is required.
For sites with discharge values or floodplain widths significantly less than the suggested ranges, Equation 6.13 (for English units) or Equation 6.14 (for SI units) should be used to compute the expansion ratio (ER):
(6.13)
and
(6.14)
The adjusted determination coefficient for Equation 6.13 and Equation 6.14 is 0.71 and the standard error of estimate is 0.26, so significant uncertainty is still present in the results of both equations. After an initial location of the sections defining the bridge (possibly based on Table 6.3 and Table 6.4), HEC-RAS output can provide the variables at sections 1 and 2 to solve these equations. The calculated variables then allow an improved estimate for expansion reach data when substituted into Equation 6.11 (6.12) or Equation 6.13 (6.14). The initial location of the contraction and expansion sections may then be adjusted to reflect the results of the appropriate equation.
Example 6.5 Computing the expansion ratio when Equation 6.11 is not applicable.
Determine the expansion ratio (ER) for the bridge of Example 6.3.
Solution
For the bridge and reach data given, b/B = 60/220 = 0.27; the ratio of overbank to channel n was found in Example 6.3 as 1.6. With these values, Table 6.4 is used to determine a preliminary estimate of ER.
As in Example 6.4, an initial value of ER is needed so that the HEC-RAS model may be coded and operated. However, the floodplain width, bridge width, and stream slope are not within or near the range of widths for which Equation 6.11 was developed. Consequently, Equation 6.11 cannot be used to compute the expansion length for this bridge. For streams with discharge values less than those for which Equation 6.11 was derived, Equation 6.13 is used to determine ER, using model output after an initial HEC-RAS run, and then length is determined with ER and the obstruction length.
Make an initial estimate of the parameters required to determine an ER from Table 6.4. For a stream slope greater than 10 ft/mi and a ratio of overbank to channel n of 1.6, an ER is interpolated from Table 6.4. For higher discharges, an ER of 2 is appropriate for a stream slope of 10 ft/mi. A lower ER can probably be used, as ER appears to decrease as slope increases. However, select an ER of 2 for a conservative initial estimate. With the obstruction length of 80 ft found in Example 6.3, the first estimate of the expansion distance is 2 × 80 = 160 ft, which is then coded to the HEC-RAS model. The HEC-RAS detailed output at sections 1 and 2, just downstream of the bridge, are reviewed to obtain the additional parameters required to solve Equation 6.13. The parameters for channel Froude Numbers at sections 1 and 2 were found in Example 6.3.
Applying Equation 6.13, the expansion ratio is found to be
The revised expansion length is 1.59 × 80 = 127 ft, or 33 ft (21 percent) less than initially estimated. The distance from the downstream bridge face to the end of expansion (section 1) is adjusted to the new value of 127 ft within the data set. Any geometry changes necessary to reflect the new location are made and the program is rerun. Any significant revision in the channel Froude Numbers at sections 1 and 2 downstream of the bridge should be reapplied to Equation 6.13, then ER and the expansion length recomputed and the program rerun until no significant changes in ER occur. The value computed with Equation 6.13 should be reasonable, with a range of allowable ER between 0.5 and 4.
When discharges are significantly greater than 30,000 ft3/s (850 m3/s), the location of the end of expansion may be overestimated by Equation 6.11 or Equation 6.13. For these discharge conditions, the following equation is more appropriate:
(6.15)
The adjusted determination coefficient for this equation is 0.59 with a standard error of 0.31, indicating significant uncertainty in the results.
Example 6.6 Computing the expansion ratio for discharges greatly exceeding 30,000 ft3/s.
Determine the expansion ratio (ER) at a bridge site having the following parameters:
Solution
B and b far exceed the maximum values for Equation 6.11 and the discharge greatly exceeds the 30,000 ft3/s upper limit. Table 6.4 is not applicable for values significantly outside the range shown. However, the upper limit for ER for stream slopes similar to the example is approximately 2. Therefore, initial estimates of ER (and CR, Cc, and Ce) are made and the bridge reach is modeled in HEC-RAS. The output from the initial run is inspected and the values of the channel Froude Numbers at sections 1 and 2 are found as 0.16 and 0.25, respectively. These values are then substituted in Equation 6.15 to give
The revised expansion length is then determined with the new, computed value of ER and the program is rerun to evaluate the need for any further adjustments in ER and Le.
The expansion reach length is derived from the most appropriate of the three equations, with the calculated value checked against the range of values in Table 6.4 for appropriateness. If the ER is greater than 3, the modeler should consider using intermediate sections between sections 1 and 2 to more accurately compute the average friction slope between the two sections. A computed ER greater than 4 should be adjusted downward to a more reasonable value.
As with the CR and contraction lengths discussed in the previous paragraphs, the ER and expansion length are based on a single discharge value, theoretically requiring different geometry sets for each profile analyzed. The development of an appropriate ER or expansion length for each bridge is handled similarly to the method for contraction length. A single, representative ER or expansion length is normally chosen, based on an "average" flood discharge or the largest flood that does not greatly overtop the bridge or embankments and this value is used for all further calculations.
Sensitivity tests can be performed to assess the effect of varying the ER or expansion reach length, based on various discharges. The traditional 1:4 ER overestimates the losses between the end of the expansion location and the downstream bridge face, giving a higher water surface and a lower velocity estimate. While this might be considered a factor of safety, these values result in less accurate profiles. Lower velocity estimates can result in underestimating potential scour, which may ultimately result in safety and stability issues for the bridge structure and foundation. Therefore, there is little justification in using the traditional 1:4 ER. Similarly, the USGS method of locating the end of expansion a distance equal to the bridge opening width was not validated by the HEC study.
Loss Coefficients for Flow Through Bridges
Expansion and contraction losses are defined in Chapter 5. These losses are generally more severe (higher) through a bridge than a natural valley cross section, due to the structure's constriction of the flow. Contraction and expansion loss coefficients of 0.3 and 0.5, respectively, are widely used in bridge analyses in a subcritical flow regime. However, the same tests that developed prediction equations for CR and ER also examined expansion and contraction coefficients. This study found that the traditional values of the loss coefficients generally overestimate the actual value of the expansion and contraction losses through the bridge reach. Manning's n is another coefficient that may require modification at bridges.
Contraction Coefficient.
The contraction coefficient (Cc) is applied to the absolute difference in velocity heads at adjacent cross sections approaching and entering a bridge. The contraction coefficient is applied when the velocity head at the downstream cross section is greater than the velocity head at the upstream cross section. In this situation, the flow area contracts. For subcritical flow, HEC-RAS normally applies the contraction coefficient at sections 4, 3, BU, and usually BD. For the tests performed in the HEC research, nearly all the profiles through bridges were best modeled with a Cc value of 0.1. For this reason, 0.1 is the accepted minimum value assigned for subcritical flow conditions through bridges. Values for all tests ranged from 0.1 to 0.5, with an average of 0.12. Due to the narrow range of results for contraction coefficients, the research suggests that Table 6.5 be employed to estimate Cc.
|
Degree of Constriction
|
Recommended Cc
|
|---|---|
|
0% < b/B < 25%
|
0.3-0.5
|
|
25% < b/B < 50%
|
0.1-0.3
|
|
50% < b/B < 100%
|
0.1
|
In Table 6.5, b is the width of the bridge opening and B is the width of the floodplain flow. The selection of B should be based on the largest discharge that does not overtop the bridge or the approach embankments. When the roadway is overtopped and a significant amount of flood flow passes over the obstruction, the contraction and/or expansion will not necessarily be as significant as when all flow is confined to pass through the bridge opening.
As Table 6.5 suggests, for most situations the traditional value of Cc = 0.3 at bridges is conservative, except for narrow bridge openings (b/B < 25%) and a value of 0.1 for Cc is now applicable for many bridges. However, the contraction coefficient used between natural channel cross sections (without a bridge) is also 0.1 and it would seem that a Cc for a cross section representing a bridge should be larger than for nonbridge sections. Without any actual data for calibration at a bridge, the modeler may opt to be conservative and retain the traditional value of 0.3 for bridges. No useful regression equations have been developed for the contraction coefficient.
Expansion Coefficient.
The expansion coefficient, Ce, is applied to the absolute difference in velocity heads at adjacent sections leaving the bridge. For subcritical flow, the expansion coefficient is used when the downstream section's velocity head is less than the upstream section's, indicating decreasing velocity and thus an expansion of flow area. In subcritical flow computations, the expansion coefficient is applied to section 2 and occasionally to section BD. Because the water surface elevation at section 1 is computed from downstream conditions, the normal valley contraction and expansion coefficients (0.1/0.3) are used at section 1, rather than the bridge expansion and contraction coefficients. As with the contraction coefficient, regression analysis did not provide a strong statistical relationship to develop an equation for the expansion coefficient. The expansion coefficients that best reproduced the known water surface profiles in the HEC tests ranged from 0.1 to 0.65, with an average value of 0.3. The only equation that showed a reasonable correlation with the hydraulic variables is
(6.16)
A corresponding equation for SI units was not developed, although the modeler may convert metric hydraulic depths to their English unit values to apply Equation 6.16. The adjusted determination coefficient for this equation was 0.55 with a standard error of estimate of 0.10, which indicates a high level of uncertainty in the prediction of Ce. Hydraulic depth for the channel and overbank and Froude Numbers for the channel at sections 1 and 2 can be obtained from the detailed section printout. These are used to get an initial estimate of the location of section 1 and in Equation 6.16 to estimate Ce.
Example 6.7 Computing the expansion and contraction coefficients for the bridge of Example 6.2.
The bridge crossing of Example 6.2 has the following parameters, which are needed for the initial estimate of the contraction and expansion coefficients:
Solution
The contraction coefficient value is estimated using Table 6.5, knowing the bridge-opening width ratio. The range of Cc from Table 6.5 for a bridge ratio of 400/1200 = 0.33 is 0.1-0.3. In general, the lower the ratio, the higher the expected value of Cc. The modeler is free to choose any value of Cc within this range. For the initial estimate before computing Ce, assume Cc = 0.2.
The expansion coefficient is estimated using Equation 6.16. However, HEC-RAS must be run to obtain the hydraulic depth and Froude numbers at sections 1 and 2, downstream of the bridge, before applying Equation 6.16. Since Cc = 0.2 was selected (midrange of contraction coefficient values), an initial selection of Ce based on the mid-range (0.3-0.5) can be assumed appropriate, or Ce = 0.4. These values are then included for the bridge reach (upstream of section 1 through section 4), along with the initial estimates for contraction and expansion length, and the HEC-RAS model is operated. From Example 6.2, the channel Froude Numbers at sections 1 and 2 are 0.21 and 0.52, respectively. The hydraulic depth for the channel at section 1 is obtained from the detailed cross-section output (D = 18.6 ft). The hydraulic depth for the overbanks at section 1 may be computed by the modeler from the top width of the section, subtracting the channel width and adding the cross-sectional area of the right and left overbank areas. The hydraulic depth of the overbank is then found by dividing overbank area by overbank top width, for a value of 4.6 ft. Hydraulic depth, left and hydraulic depth, right are also available as HEC-RAS variables.
With the values generated by HEC-RAS, Equation 6.16 gives the estimate of Ce as
The computed value for Ce is about 60 percent of the initial estimate and nearly the same as the initial estimate for Cc. Because Cc is typically much less than Ce, the modeler may consider decreasing Cc to reflect the computed value of Ce. A simple proportion can be used to give
HEC-RAS should be rerun using the new coefficients and the revised value of channel Froude Number at section 2 evaluated. If the revised value is significantly different than the initial value of 0.52, Equation 6.16 should be reapplied to determine a revised value of Ce. The computed values for the two coefficients are within the allowable range of possible values; however, the computed value of Ce for the bridge is less than that for nonbridge sections. The modeler must determine whether this represents a realistic value or whether a minimum value of 0.3 should be used for the bridge reach.
Example 6.8 Computing the expansion and contraction coefficients for the bridge of Example 6.3.
For the bridge crossing of Example 6.3, the following parameters are needed for the initial estimate of the coefficients:
Solution
For b/B between 0.25 and 0.5, the contraction coefficient ranges from 0.1 to 0.3, as shown in Table 6.5. Since the actual bridge opening ratio is close to 0.25, an initial estimate of the contraction coefficient of 0.3 is appropriate (smaller ratio, higher coefficient). However, the modeler may select a different value within this range, if desired.
Because the value of Cc initially adopted reflects the "traditional" value for bridges, an initial value of Ce = 0.5 is also used. These values are then included for the bridge reach (upstream of section 1 through section 4), along with the initial estimates for contraction and expansion length, and HEC-RAS is rerun. From Example 6.3, the channel Froude Numbers at sections 1 and 2 are 0.3 and 0.64, respectively. The hydraulic depth for the channel at section 1 is obtained from the detailed section output (D = 9.4 ft). The hydraulic depth for the overbank at section 1 may be computed from the top width of the full cross section, subtracting the channel width and adding the cross-sectional area of the right and left overbank areas. The hydraulic depth of the overbank is then found by dividing overbank area by overbank top width, for a value of 4.3 ft. Hydraulic depth, left and hydraulic depth, right are also available as HEC-RAS variables.
With these values from the HEC-RAS output, Equation 6.14 estimates Ce as
.
Because the computed value for Ce is much lower than the initial value, it is reasonable to adjust the initial estimate for Cc in proportion to the change in the expansion coefficient. Because Cc is typically much less than Ce, the modeler may consider decreasing Cc to reflect the computed value of Ce. A simple proportion can be used to give
.
HEC-RAS is then run again with the new coefficients and the revised value of channel Froude Number at section 2 evaluated. If the revised value is significantly different from the initial value of 0.64, Equation 6.16 is reapplied in order to determine a revised value of Ce. Both the coefficient values exceed those for nonbridge sections, which appears reasonable.
If there is a long contraction or expansion reach at the bridge, resulting in large differences in conveyance and friction slope between sections 1 and 2, the modeler should insert intermediate sections. These intermediate sections reflect the higher coefficients used to model bridges. Intermediate sections should also include the ineffective flow option used at bridges, presented in detail later in this chapter.
The expansion coefficient is more important than the contraction coefficient when analyzing bridge losses. This is because more energy is lost in an expansion than in a contraction. The photograph in Figure 6.8 illustrates this phenomenon, where one can observe the smooth streamlines entering the bridge, compared to the high turbulence leaving the bridge. For this reason, sensitivity tests on the selected expansion coefficient should be considered. HEC's guidance suggests operating the model with expansion coefficient values of ±0.2 from the computed value. This increment represents ±2 standard deviations of a 95-percent confidence band around the computed value. If the difference in the water surface elevations through the bridge is large for the range of Ce, a conservative (high) value of Ce is warranted. Where the added increment gives a very large value for the coefficient, an upper limit for Ce of 0.8, equal to an abrupt expansion, is recommended. Where the subtracted increment gives values of Ce less than 0.1, a minimum value of 0.1 is recommended.
The traditional expansion and contraction values through bridges generally result in a conservative estimate of the nonfriction losses. This provides a factor of safety when computing bridge losses. However, the higher expansion coefficients result in higher water surface elevations and, therefore, less accurate water surface profiles at the bridge face. The higher water surface elevations at the bridge face produce a lower velocity, which can cause errors in bridge scour computations. With the work done by HEC, lower values of expansion and contraction coefficients may be more representative of field conditions and are based on a scientific study calibrating Cc and Ce against measured discharge and highwater mark data. The traditional coefficients are found to be appropriate only for those bridge openings that represent less than 25 percent of the effective flow width in the floodplain. However, the modeler must make the final selection of the expansion and contraction coefficients. Without detailed data to calibrate a profile through a bridge, the modeler may opt for retaining the more traditional, and generally higher, values of Cc and Ce.
Loss Coefficients for Supercritical Flow.
No firm guidance is available for bridge coefficient selection in a supercritical flow regime. Although the same values for expansion and contraction coefficients may be used as described in the previous section for subcritical flow, lower values are normally more appropriate for modeling supercritical flow reaches. The guidance given in Chapter 5 for supercritical flow, in normal valley cross sections, suggests that values of 0.05 and 0.1 for the contraction and expansion coefficients, respectively, represent the upper limits (to avoid oscillation of the computed profile).
Because the profile computations proceed upstream to downstream in supercritical flow, section 4 has expansion and contraction coefficients similar to a normal valley section. Sections 3, BU, and usually BD are expected to represent expansions, because obstacles in supercritical flow cause the flow velocity to decrease and the water surface elevation to increase. Thus, Ce is applied by HEC-RAS to sections 3, BU, and usually BD. Sections 1, 2, and occasionally BD are expected to represent contractions in supercritical flow, since the velocity increases after passing the obstruction and the water surface elevation decreases. Thus, Cc would be applied by HEC-RAS at sections 1, 2, and occasionally BD. In natural channels experiencing supercritical flow, the guidance given by HEC-RAS suggests lower expansion and contraction coefficients than for the same bridge under subcritical flow. Suggested values for most bridges for supercritical contraction and expansion coefficients are 0.1 and 0.3, respectively, and 0.3 and 0.5, respectively, at very abrupt transitions. However, most instances of supercritical flow over a length of stream containing a bridge occur within man-made channels of nearly constant cross-sectional area. Where these nearprismatic channels are present, expansion and contraction values of zero or near zero (0.01 to 0.02) are often used. Physical model testing may ultimately be required to properly and adequately address the effect on water surface profiles for bridge design under supercritical flow conditions.
Manning's n at Bridges.
The channel and floodplain Manning's n values can require adjustment at bridge cross sections. Sections 2 and 3 are located a short distance from the bridge face and may have a lower value of Manning's n for both the channel and floodplain, compared to sections 1 and 4. Normal road maintenance includes periodically mowing the bridge right-of-way and removing tree growth for several yards (meters) on either side of the embankment toe, where sections 2 and 3 are often located. Reduced vegetation along the alignment of these two sections can result in an overbank n value representative of grass rather than the dense brush or woods which may exist upstream or downstream of the bridge. The modeler will have to estimate an appropriate average n value for sections 2 and 3, depending on the flow regime, possibly based on a distance weighting of n, because these sections should be representative of the roughness for one-half the distance to the next cross section (1 or 4). Similarly, the original bridge design or later erosion problems may have resulted in a widened, straightened, or lined (with concrete or rock, for instance) channel through the bridge, compared to the original channel configuration and roughness. This situation again results in a lower value of n for the channel at sections 2 and 3, compared to those of sections 1 and 4. The modeler may want to apply lower values of Manning's n at sections 2 and 3 to more accurately represent actual conditions. If the channel through the bridge is much larger than the channel at sections 1 or 4, a transition section can be considered between 1 and 2 or between 3 and 4, where the change in channel cross section (and channel and overbank n) can be specified.
Cross sections BD and BU, automatically added by HEC-RAS, use the Manning's n values associated with sections 2 and 3. However, the modeler may overwrite the Manning's n values for BD and BU to reflect a different condition within the bridge. For instance, if the channel is paved inside the bridge but not at sections 2 and 3, the n value for concrete can be substituted for the n value at sections BD and BU. The channel and floodplain geometry inside the bridge opening for sections BD and BU can also be adjusted, if needed.
Similarly, if the profile analyses for discharges that overtop the roadway are being computed using the energy method, the Manning's n reflecting the roadway surface should be used for sections BD and BU to reflect the roadway surface at these sections. Because of the normally short length of a bridge, friction losses computed with Manning's n are usually small. Other losses (expansion and contraction) are typically much larger and, therefore, more critical than the friction losses through the bridge.
The engineer should check the conveyance and friction slope values between sections 1 and 2 and between 3 and 4 for large differences that may in turn lead to large differences in water surface elevations. If there are large changes in water surface elevations between these cross sections, intermediate sections may be needed to best model the change in n between each pair of sections.
