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Publication Number: FHWA-HRT-09-028
Date: May 2009

 

Hydrodynamic Forces on Inundated Bridge Decks

5. Deck force calculation examples

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 six-girder 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 50-year flood rises to an elevation of 105.9 m and is passed by the bridge. The 100-year flood has an elevation of 107.5 m and has an average stream velocity of 2.3 m/s. The 1,000-year flood rises to elevation 110 m and has an average velocity of 3.2 m/s.

Using these dimensions, hu, hd, h*, and Fr can be calculated. Table 5 calculates these flow conditions for the two floods.

Table 5. Flow conditions for example design floods.

100-year flood

1,000-year flood

hu = 107.5 - 102 = 5.5 m

hu = 10 m

hb = 106 - 102 = 4 m

hb = 4 m

h* = (hu - hd)/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 six-girder 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 100-year flood, take the equation in figure 37 and apply the six-girder, high estimate coefficients to get CD as a function of h* as in figure 42.

Figure 42. Equation. Upper fitting equation for drag coefficient as a function of h*. C subscript D as a function of h (star) equals the sum of the product 2.7 times base e raised to the product of -2 times h (star) squared, that product plus the product of negative 2.7 times base e raised to the product of -0.75 times h (star) squared, that product plus 2.15.

Figure 42. Equation. Upper fitting equation for drag coefficient as a function of h*.

For the 100-year flood, substitute h* = 0.6 into the 6-g upper equation (figure 42) to find the high estimate of CD 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

100-year high

100-year low

1,000-year high

1,000-year low

CD

1.403

1.003

2.114

1.714

CL

-0.955

-1.462

-0.112

-0.192

CM

0.247

0.092

0.0318

-0.0695

Table 6 shows that the 1,000-year flood has the higher force coefficient when the drag force is considered, but, for the lift force and moment, the 100-year 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 1,000-year 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 six-girder bridge for the 1,000-year flood. F sub D divided by L equals the product C subscript D times one-half times rho times v squared times s equals the product of 2.114 times one-half times 1,000 times 3.2 squared times 2.5, which equals 27.1 kilonewtons per meter.

Figure 43. Equation. Total drag force per unit length on the example six-girder bridge for the 1,000-year 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. Velocity, v, at h*crit. v equals the product of Fr times the square root of the product g times the sum of the product h (star) subscript crit times s, that product plus h subscript b equals 0.323 times the square root of the product 9.8 multiplied by the sum of the product 3 times 2.5, that product plus 4 equals 3.43 meters per second.

Figure 44. Equation. Velocity, v, at h*crit.

Using the velocity from figure 44 and the critical value of CD (2.15), the total drag force is 31.6 kN/m. Now, suppose a streamlined bridge is considered instead of the six-girder bridge. While L, W, and hb 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 CD 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 six-girder bridge. Similar calculations may be followed to analyze the lift force and moment.


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The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT). The hydraulics and hydrology research program at the TFHRC Federal Highway Administration's (FHWA) R&T Web site portal, which provides access to or information about the Agency’s R&T program, projects, partnerships, publications, and results.
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