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Publication Number:
FHWAHRT09028
Date: May 2009 
This section describes how to use the results of this study to estimate the force and moment coefficients on a proposed bridge design.
To calculate the force coefficient, the dimensions and elevation of the proposed bridge, the dimensions of the channel, and the velocity and inundation elevation of the stream at the flood of interest must be known.
Suppose a small sixgirder bridge is planned with dimensions as in table 4.
Table 4. Bridge example dimensions.
Dimension 
Value 

Deck thickness (s) 
2.5 m 
Length (L) 
280 m 
Width (W) 
11.2 m 
Low chord elevation 
106 m (NAVD88) 
The bridge is planned over a small stream with an enormous range of flow. Suppose that the stream has a channel bottom elevation of 102 m at the crossing. The 50year flood rises to an elevation of 105.9 m and is passed by the bridge. The 100year flood has an elevation of 107.5 m and has an average stream velocity of 2.3 m/s. The 1,000year flood rises to elevation 110 m and has an average velocity of 3.2 m/s.
Using these dimensions, h_{u}, h_{d}, h*, and Fr can be calculated. Table 5 calculates these flow conditions for the two floods.
Table 5. Flow conditions for example design floods.
100year flood 
1,000year flood 

h_{u} = 107.5  102 = 5.5 m 
h_{u} = 10 m 
h_{b} = 106  102 = 4 m 
h_{b} = 4 m 
h* = (h_{u}  h_{d})/s = (5.5  4)/2.5 = 0.6 
h* = 2.4 
Fr = 2.3/(9.8 x 5.5) _{0.5} = 0.313 
Fr = 0.323 
Supposing the bridge is a sixgirder bridge deck, the force coefficients can be calculated for the two floods using the equations in figure 37 to figure 39 and the appropriate coefficients in table 3.
For example, to compute the drag coefficient for the 100year flood, take the equation in figure 37 and apply the sixgirder, high estimate coefficients to get C_{D} as a function of h* as in figure 42.
Figure 42. Equation. Upper fitting equation for drag coefficient as a function of h*.
For the 100year flood, substitute h* = 0.6 into the 6g upper equation (figure 42) to find the high estimate of C_{D} equals 1.403. A similar process is followed to get the force coefficient values in table 6 for both example floods.
Table 6. High and low force coefficients for the two example floods.
Force coefficient 
100year high 
100year low 
1,000year high 
1,000year low 

C_{D} 
1.403 
1.003 
2.114 
1.714 
C_{L} 
0.955 
1.462 
0.112 
0.192 
C_{M} 
0.247 
0.092 
0.0318 
0.0695 
Table 6 shows that the 1,000year flood has the higher force coefficient when the drag force is considered, but, for the lift force and moment, the 100year flood values are more important. It should be noted that the h*_{crit} value occurs between the two floods for the lift and moment coefficients, so for design purposes, it is preferable to use the critical coefficient values in table 3.
Figure 5 through figure 7 allow the total forces and moments to be calculated. For instance, figure 5 can be rewritten to express the total drag force per unit length, as shown in figure 43, which inserts the bridge dimensions and flow values for the 1,000year flood and solves the force as approximately 27.1 kilonewtons per meter (kN/m).
Figure 43. Equation. Total drag force per unit length on the example sixgirder bridge for the 1,000year flood.
The designer may calculate the maximum drag force by knowing the velocity at the critical value of h*, which can be computed if the Fr is assumed to be constant. By combining figure 2 and figure 3 and keeping the Fr constant, velocity equals 3.43 m/s (at h* = 3), as shown in figure 44.
Figure 44. Equation. Velocity, v, at h*_{crit}.
Using the velocity from figure 44 and the critical value of C_{D} (2.15), the total drag force is 31.6 kN/m. Now, suppose a streamlined bridge is considered instead of the sixgirder bridge. While L, W, and h_{b} remain the same, the bridge thickness is reduced by the same proportion as the experimental prototypes to s = 1.64 m. The critical value of C_{D} for the streamlined bridge is 1.1 at roughly h* = 5, or 4.57 if the velocity remains the same. Solving the equation in figure 43 with the streamlined parameters, the total drag force is 11.2 kN/m, which is a sizeable reduction from the sixgirder bridge. Similar calculations may be followed to analyze the lift force and moment.
Topics: research, infrastructure, structures, bridge hydraulics, scour Keywords: Bridge decks, bridge design, bridge foundations, bridge hydraulics, bridge inundation, bridge scour, pressure flows, pressure scour, submerged flows Updated: 07/10/2012
