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

 

Hydrodynamic Forces on Inundated Bridge Decks

2. Theory and Approach

The study for this report investigated the forces acting on inundated bridge decks through physical experimental testing and CFD modeling with two commercial software packages. Three types of bridges were tested for their reaction to drag forces, lift forces, and moment-generating forces in relation to the degree of inundation (inundation ratio). The bridge types included traditional six-girder and three-girder shapes, and prototype streamlined deck shapes designed to reduce the force load during inundation.

The first technical task performed for this study was the analysis of drag and lift forces and moments for inundated bridge decks as a function of immersion height and flow rate. The analysis results for these prototypical open channel flow configurations were compared to the force balance and Particle Image Velocimetry (PIV) measurements from TFHRC tests to validate the corresponding CFD models. Both two-dimensional (2-D) and three-dimensional (3-D) models of the bridge decks with six- and three-girders were studied using Fluent® and STAR-CD® CFD software. A set of simulations were completed to assess the impact of various modeling options, including the following, while always considering accuracy of results and computational efficiency:

Theory of Forces Acting on Inundated Bridge Decks

In determining the structural response of the bridge decks, both the properties of the fluid and the configuration of the bridge were important. Consistent with the experimental methodology, the nomenclature for bridge and flume dimensions are illustrated in figure 1.

Figure 1. Diagram. Scaled six-girder bridge deck model used in assessments of drag and lift forces and the nomenclature for bridge deck and flume dimensions. This figure depicts the cross section of a bridge deck with six girders submerged in a fluid characterized by a free surface indicator at the top and black line at the bottom. The total depth from the free surface to the bottom is labeled as h subscript u. The height from the bottom of the fluid to the bottom of the bridge decks girders is labeled as h subscript b. The width of the bridge deck has dimension, W. The face of the bridge deck from the bottom of the girder to the top of side wall is s. A horizontal vector pointing from left to right indicates direction of flow and is labeled V for velocity.

Figure 1. Diagram. Scaled six-girder bridge deck model used in assessments of drag and lift forces and the nomenclature for bridge deck and flume dimensions.

In determining the forces on bridges, the properties of the flowing stream that had the greatest impact were the height of the water and its velocity. The height of the water surface in relation to the flume dimension, the position of the bridge deck, as well as the flow velocity can be expressed as the inundation ratio, h*. Shown in figure 2, the inundation ratio is a measure of the difference between the height of the free surface over the low chord of the bridge deck (hu - hb) divided by the deck thickness, s. Higher values of h* mean that the bridge is more submerged. The flow velocity can be characterized by the dimensionless Fr. Fr (figure 3) is expressed as the free stream velocity, v, over the square root of the depth of flow multiplied by the gravitational acceleration, g

Figure 2. Equation. Inundation ratio. h (star) equals the difference of h subscript u minus h subscript b, divided by s.

Figure 2. Equation. Inundation ratio.

Figure 3. Equation. Froude number. Fr equals v divided by the square root of the product of g times h subscript u.

Figure 3. Equation. Froude number.

There were three primary responses of the bridge deck to the flowing water. Figure 4 illustrates the forces acting on a submerged bridge deck. The drag force, FD, acting parallel to flow, affects the bridge deck horizontal stability and tends to "push" the bridge off its piers and abutments. The lift force, FL, acts vertically and perpendicular to flow and can lift the bridge deck off of its piers and abutments. Unevenly distributed forces on the bridge decks lead to moments about the center of gravity, Mcg, of the bridge deck, which can cause the bridge to overturn.

Figure 4. Diagram. Definition sketch of forces acting on bridge deck. A six-girder bridge deck in a cross section is shown with height, s; width, W; and center of gravity marked as a point labeled cg. A flow vector, V, approaches the bridge from the left and points right. The drag force vector, F sub D, points right from cg. The lift force vector, F sub L, points upward from cg. The moment vector, M sub cg, is shown as an arc pointing counterclockwise about cg. All three force vectors are shown in their positive directions.

Figure 4. Diagram. Definition sketch of forces acting on bridge deck.

The forces of drag and lift and moments acting on the bridge deck are usually expressed in nondimensional coefficient forms. The definition of the coefficient of drag, CD, depends on h* and is shown in figure 5. The two cases correspond to a partially inundated bridge (h* < 1) and a fully submerged bridge (h* > 1).

Figure 5. Equation. Drag coefficient. This equation has two cases. The first case is for when h (star) is greater than 1. C sub D equals F sub D divided by the product of one-half times rho times v squared times L times s, end of product., if h* > 1; and The second case is for when h (star) is less than 1. C sub D equals F sub D divided by the product of one-half times rho times v squared times L times the difference h sub u minus h sub b, end of difference, end of product., if h* < 1

Figure 5. Equation. Drag coefficient.

The lift coefficient is defined in figure 6. The moment coefficient with respect to the center of gravity of the bridge deck is defined in figure 7.

Figure 6. Equation. Lift coefficient. C sub L equals F sub L divided by the product of one-half times rho times v squared times L times W, end of product.

Figure 6. Equation. Lift coefficient.

Figure 7. Equation. Moment coefficient. C sub M equals M sub cg divided by the product of one-half times rho times v squared times L multiplied by W squared, end of product.

Figure 7. Equation. Moment coefficient.

Here, FD and FL are the forces integrated over the surface of the bridge deck along the flow and perpendicular directions, respectively, where ρ is the density of water, and L is the length of the bridge (orthogonal to the plane of figure 1). When calculating the integrated vertical force, FL, its component associated with buoyancy force is excluded to be consistent with the experiments in which force balances were calibrated for zero lift under no-flow conditions.

Hydraulic Force Measurement Experimental Setup

The physical experimental setup was designed to empirically determine the effect of the inundation ratio on the lift, drag, and moment forces for three different bridge prototypes.

The experiments were conducted in a 12.8 m long by 0.4 m wide by 0.5 m high Plexiglas® rectangular flume. The flume was set horizontally, and the depth of flow was controlled with an automatic, adjustable tailgate located at the downstream end of the flume. Flow was supplied by a 0.054-m3/s pumping system. The mean free-surface level was measured using ultrasonic sensors at two cross sections along the flume. An electromagnetic flow meter measured the discharge. An Acoustic Doppler Velocimeter (ADV) probe was used to measure the velocity distribution of the flow.

The following three models of bridge decks were tested in the experiments:

Figure 8. Diagram. Dimensions of the six-girder bridge deck. A bridge deck with six thin, I-shaped, evenly spaced girders is shown in a cross section. The bridge deck is 0.058 m high and 0.26 m wide. The road surface is slightly above the midpoint of the height, and two railings, one on each side, make up the rest of the height.

Figure 8. Diagram. Dimensions of the six-girder bridge deck.

Figure 9. Diagram. Dimensions of the three-girder bridge deck. A bridge deck with three thick, rectangular, evenly spaced girders is shown in a cross section. The bridge deck is 0.070 m high and 0.26 m wide. The road surface is somewhat above the midpoint of the height, and two railings, one on each side, make up the rest of the height.

Figure 9. Diagram. Dimensions of the three-girder bridge deck.

Figure 10. Diagram. Dimensions of the streamlined bridge deck. A streamlined bridge deck is shown in a cross section with an overall width of 0.26 m and a height of 0.038 m. The leading and trailing edges come to a point in a roughly equilateral triangular shape with the interior side being vertical. The bridge deck road surface is recessed such that it nearly lines up horizontally with the pointed leading and trailing edges. This surface falls approximately 20 percent below the highpoints of the bridge. The bottom faces of the triangular edges continue diagonally downward and inward and smoothly transition to two convex circular arc lobes whose low point occurs horizontally roughly midway between the bridge edge and center point. The low point of the lobes is also the low point of the bridge. The circular arcs then continue upward and inward to a concave, circular arc of approximately threefold larger radius. The high point of the concave occurs at the horizontal and vertical midpoints of the bridge dimensions.

Figure 10. Diagram. Dimensions of the streamlined bridge deck.

Both the six-girder and three-girder bridge decks had railings on both sides with the dimensions shown in figure 11.

Figure 11. Diagram. Side view of railings with dimensions for the six-girder and three-girder bridge decks. The railing is shown from a side elevation view. The railing has a thick, solid bar at the bottom where it attaches to the side of the bridge decking. A thin, horizontal railing bar is connected to the bottom bar with posts of approximately the same height as the bridge deck thickness. The posts are 11 mm wide with a 50 mm gap between posts.

Figure 11. Diagram. Side view of railings with dimensions for the six-girder and three-girder bridge decks.

The six-girder bridge, representing a typical U.S. highway bridge deck shape, was constructed out of Polyvinyl Chloride (PVC) at a 1:40 geometrical reduction scale based on the depths, maximum discharges, and inundations possible in the flume. This scaled bridge deck was optimally sized to investigate a range of force values in flow conditions ranging from low to high flows, all at subcritical Fr in the upstream flow. The six-girder and three-girder bridge decks had the same width, but the three-girder deck had a 22 percent larger frontal area (s = 0.70 m).

The three bridges were evaluated in the same flume and under the same experimental conditions in order to reduce the experimental error. The experiments were conducted for four approach velocities ranging from 0.25 to 0.5 m/s. The flow depths, hu, in the experiments were always kept at 0.25 m. Under these conditions, the Fr varied within the range of 0.16 to 0.32 in response to the changing velocities. The submergence of the bridge deck within the 0.25-m flow depth was varied from slight submergence of the deck girders to complete overtopping of the bridge deck, corresponding to an inundation ratio of h* = 0.29 to roughly 5.0 (for the streamlined bridge).

The measurement of forces on the bridge deck was performed by a deck force analyzer system, which was constructed for this study (figure 12 and figure 13). This transducer system was designed to measure forces induced by the bridge deck model simultaneously in the x-direction (Drag) and z-direction (Lift) by use of electric strain gauges. The bridge deck model was mounted on a bracket and positioned in the flume. The bracket was attached at the platform of the system by four ball-beared pendulums which were connected with the platform by four flat springs. This allowed it to move back and forth in the direction of flow as well as vertically. Motion in each direction was resisted by two pairs of flat springs. The tension in these springs was detected by strain gauges, which generated an electrical signal proportional to the forces upon the bridge deck model in the x-direction (Drag) and z-direction (Lift).

Figure 12. Diagram. Test section of deck force analyzer showing strain gauge configuration. A side 2-D view of the test section of the deck force analyzer is shown in gray with a white background. The bottom half of the analyzer is an inverted trapezoid shape. Two rows of holes run vertically down the center of the trapezoid shape. The six-girder bridge deck is shown in a cross section. Its deck is mounted to the analyzer by the bottom two holes, which connect to the bridge on the road surface just outside the middle two girders. A mostly rectangular section about half the height of the trapezoid is above the trapezoid. At the top, left corner of this, the strain-gauge for drag connects and is labeled. It is 382 mm above the bridge roadway. Floating above the corners rectangular section are two brackets. The left one is marked as "STRAIN GAUGE-LIFT FRONT," and the right as "STRAIN GAUGE-LIFT REAR." They are 537 mm vertically above the bridge deck with 260 mm between the two.

Figure 12. Diagram. Test section of deck force analyzer showing strain gauge configuration.

Figure 13. Photo. Deck force analyzer system at the TFHRC hydraulics lab. This vertical format photograph shows the deck force analyzer as mounted in a flume filled halfway with water. The flume crosses the bottom of the picture horizontally. A tall frame, roughly square in plan view, straddles the flume. The test section is connected to the top of the frame by long, straight pendulums. The test section has two identical sections of the type shown in figure 12 sitting along the side walls of the flume. The flume is roughly half filled with water, and the test section holds the streamlined bridge deck just above the water level.

Figure 13. Photo. Deck force analyzer system at the TFHRC hydraulics lab.

The transducer was dead-load calibrated prior to use and was checked periodically during data collection. The calibration was performed by running an experiment with still water for each inundation ratio h*. The output of the acquisition data was set as the zero-reference value. The mean drag and lift forces of an experimental run were evaluated by subtracting the corresponding zero-reference value.

PIV Setup

A secondary experimental setup was built to examine the flow field around the bridges in detail using PIV, which is a noninvasive measurement technique to visualize flow distributions. The technique involves adding microscopically small particles that are highly reflective to the flow and then using a laser to illuminate a thin layer of the flow so that only the particles in that light sheet reflected the laser's light. Using cameras pointed with different angles toward the light sheet, researchers can capture images of the moving particles. In order to measure the velocity, at least two exposures were needed. They could be recorded on one or several frames. The frames were split in a large number of interrogation areas, or tiles. It was then possible to calculate a displacement vector for each tile with the help of signal processing (an auto correlation/cross correlation algorithm). This was converted to a velocity using the time between image exposures.

The PIV experiment was conducted in a smaller flume with bridge models scaled down by a factor of 1.5. The experiments were conducted for V = 250 mm/s with hb = 50 mm, 101 mm, and 116 mm and the depth of the flow, hu, as 170 mm.

Computing and TRACC

Besides establishing credible experimental data on the response of bridge decks to various forces, a major objective of this study was to establish validated computational practices to address the transportation community's research needs in bridge hydraulics. Traditionally, bridge hydraulics work relies on scaled experiments to provide measurements for flow field, which is the velocity and turbulence of a fluid as functions of position and time. Now, however, parallel computers and commercially available software provide an opportunity to shift the focus of these evaluations to the CFD domain. After being validated using the data from a limited set of experiments, high-fidelity CFD simulations can be used to expand the range and scope of parametric studies. The CFD simulations may be used in the future to predict the effects of scaling by studying differences between the reduced-scale experiments and full-scale bridges.

In this study, the reduced-scale experiments conducted at the TFHRC hydraulics laboratory established the foundations of a CFD-based simulation methodology. Researchers at Argonne and TFHRC worked together to study CFD techniques for simulating open channel flow around inundated bridges. The computationally intensive CFD modeling programs require enormous computing power.

Argonne is leading the initiative to establish the high-performance computing center, Transportation Research and Analysis Computing Center (TRACC), to pursue research and development (R&D) programs. TRACC is a general purpose advanced computing and visualization facility available to the transportation community for a broad spectrum of applications. The TRACC components include high-performance computing, visualization, and networking systems.

Argonne analysts provided technical support to researchers at TFHRC on CFD simulations for this study. Two CFD software packages were investigated for their utility in modeling open channel flow around bridges for this study. The STAR-CD® and Fluent® programs were used in this study to build models of the six- and three-girder bridges and to run simulations. The models were calibrated using the results of the six-girder reduced scale experiments and PIV results. The three-girder bridge deck was simulated using the same calibration settings as the six-girder models to serve as a "blind-test" of validity. The following chapter describes the two models and their numerical configuration and evaluation for this study.


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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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