Computers & Fluids 39 (2010) 1390–1400

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Computers & Fluids

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Effects of submersion depth on wave uplift force acting on Biloxi Bay Bridgedecks during Hurricane Katrina

Hong Xiao a, Wenrui Huang a,c,*, Qin Chen b

a Department of Civil and Environmental Engineering, FAMU-FSU College of Engineering, 2525 Pottsdamer Street, Tallahassee, FL 32310, USAb Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, Louisiana, LA 70803, USAc Tongji University, 1239 Siping Road, Shanghai, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 24 November 2008Received in revised form 16 March 2010Accepted 14 April 2010Available online 18 April 2010

Keywords:Wave load modelUplift forceBridge deckHurricane KatrinaStorm surge

0045-7930/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.compfluid.2010.04.009

* Corresponding author. Address: Department of Cneering, FAMU-FSU College of Engineering, 2525 Potts32310, USA.

E-mail address: whuang@eng.fsu.edu (W. Huang).

A large portion of the Biloxi Bay Bridge was submerged and destroyed by surface waves and storm surgeassociated with Hurricane Katrina in 2005. In this paper, the time history of wave forces exerted on theBiloxi Bay Bridge during Hurricane Katrina was investigated by a wave-loading model. The Volume ofFluid (VOF) method was adopted in the model to track the variations of water surface levels. In orderto obtain wave parameters and storm-surge elevation at the bridge site during Hurricane Katrina, a stormsurge model and a wave propagation model were coupled to hindcast the hydrodynamic conditions. Out-puts of the coupled wave–surge models were imported to the wave-loading model to simulate thedynamic wave forces acting on the bridge deck. In order to evaluate the maximum uplift wave force, fivedifferent bridge deck elevations submerged at different water depths were investigated. The processes ofwave–bridge interaction were simulated by the wave-loading model. The wave profiles, velocity field inthe vicinity of the bridge, and dynamic wave forces on the decks were analyzed. Results indicate that theuplift force on the submerged bridge deck span exceeded its own weight under the extreme wave andstorm surge conditions during Hurricane Katrina. Moreover, the numerical simulations suggest thatthe maximum uplift wave force occurred when the storm surge water level reached the top of the bridgedeck.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The Biloxi Bay Bridge is located in the state of Mississippi, USA,which carries US Route 90 over Biloxi Bay between Biloxi andOcean Springs. A large portion of the bridge was completely sub-merged underwater and severely damaged during Hurricane Kat-rina in August 2005 (Fig. 1). The Biloxi Bay Bridge was a simply-supported bridge with spans placed on pile caps. Each of the bridgespans consisted of a 0.3-m thick deck and several 1.05-m high gird-ers. The deck was cast integrally with the girders. The deck was15.85 m long and 10.19 m wide. The bridge had two spans placedside-by-side with east and westbound traffic lanes on opposingsides.

A number of investigators have studied the interaction of waveswith marine structures. El Ghamry [8] investigated vertical waveforces exerted on a horizontal deck under the action of periodicwaves using small-scale laboratory experiments. The author alsoused the linearized potential flow theory with a free surface to esti-

ll rights reserved.

ivil and Environmental Engi-damer Street, Tallahassee, FL

mate uplifting wave forces. Wang [22] developed simple rules toestimate the maximum uplift pressures on a pier deck based onthe linear wave theory and laboratory tests. It was found in [22]study that the slowly-varying pressure component ranged fromone to two times the hydrostatic pressure for a pier above a beachwith a 1:14 slope. French [9] investigated the rapidly varying peakuplift pressure and slowly varying uplift pressure on a horizontalpier with positive soffit clearance by theoretical and experimentmethods. Tanimoto and Takahashi [20] conducted laboratoryexperiments to study the horizontal and vertical components ofwave forces on a rigid platform due to periodic waves. The totaluplift pressure exerted on a horizontal platform was separated intoa shock pressure component and a static pressure component. Ira-djpanah [12] presented a finite element model to investigate thehydrodynamic effects on a horizontal platform. The flow was as-sumed to be inviscid and irrotational. Kaplan et al. [15] analyzedthe wave impact forces acting on offshore platform deck structuresin large incident waves theoretically and experimentally. Theauthors also investigated the effect of wave heading angles relativeto different structural elements. Isaacsion and Bhat [13] conductedan experimental study on the vertical force due to regular, non-breaking waves acting on a rigid horizontal plate located nearthe water surface. Tirindelli et al. [21] conducted a series of phys-

xxxxxxxx xx xxxxx x xx x

x xxxxxxxxxxxx xx

xxxxxx x xxx

Distance from West Bank of the Biloxi Bay Bridge (m)

Elevation(m,NAVD88)

0 500 1000 1500 2000 2500-5

0

5

10

15

Bay Bottom

Maximum Surge

Crest Elevation of Significant Waves

Bridge Deck Elevation

Deck Collaspe

CASE IIICASE II

CASE I

Fig. 2. Elevation view of water levels and bridge finished grade over Biloxi Bay.

Fig. 1. Photograph of US 90 Bridge over Biloxi Bay.

H. Xiao et al. / Computers & Fluids 39 (2010) 1390–1400 1391

ical model studies to measure the wave-induced loading on a mod-el of an open-piled jetty structure. These measurements gave a ser-ies of force and pressure data on selected structural elements of themodel jetty, covering a wide range of wave conditions and deckgeometries. Douglass et al. [7] provided a synthesis of existingknowledge related to hurricane wave forces on highway bridgesuperstructures. Based on pre-existing laboratory experimentaldata, Douglass et al. [7] proposed a simple empirical equation forestimating wave loads on bridge decks. In response to the devasta-tion of the 2004 and 2005 hurricane seasons, a number of newphysical experiments on wave forces on bridge decks with realisticconfiguration similar to the damaged bridges on the Gulf Coasthave been reported in the coastal engineering literature [2].

Recently, Huang and Xiao [10] used a numerical model based onthe VOF method to simulate wave forces exerted on the EscambiaBay Bridge, Florida, which was severely damaged during HurricaneIvan (2004). They have obtained and analyzed the time history ofuplifting and horizontal wave forces acting on the bridge deck un-der extreme wave and storm surge conditions. For the wave andwater elevation conditions presented in Huang and Xiao [10], thebridge deck bottom was above the maximum surge height. How-ever, in the case of Hurricane Katrina (2005), a large portion ofthe Biloxi Bay Bridge was completely submerged in the water.Chen et al. [3] analyzed wave forces on the un-submerged decksof the Biloxi Bay Bridge. Although most of the wave energy is con-centrated near the water surface, how the wave forces vary withthe depth of submergence for the fully submerged decks of the Bi-loxi Bay Bridge is unclear. Would the wave forces of submergedbridge deck be smaller than those of emergent or partially sub-merged decks? The objective of the study is to seek answers tothose questions.

In this study, numerical computations are carried out to im-prove our understanding of the wave impact on bridge decks withvarying submersion depths. A wave-loading model based on theReynolds-Averaged Navier–Stokes (RANS) equation is employed.The experimental data of uplift force on a horizontal plate [9] isused to validate the numerical model. Following the validation,the model is applied to the simulation of the dynamic wave loadsacting on the full-scaled decks of the Biloxi Bay Bridge during Hur-ricane Katrina. Both the process of wave–bridge interaction and theuplift and horizontal components of the wave force exerted onbridge decks are investigated.

Based on the numerical results, the variations of uplift forcewith the submersion depth of bridge decks are also analyzed.Fig. 2 shows the elevation view of the bay bottom, the maximumsurge water level, the crest elevation of the significant waves, thebridge finished grade, and the locations of the collapsed bridge

spans over Biloxi Bay. In order to investigate the effect of submer-sion depth on wave forces, bridge decks at three different eleva-tions related to the surge height were selected: Case I: emergent(where the deck or girder bottom is at the surge water surface),Case II: half submerged (where half of the deck or girder heightis below the surge water surface), and Case III: fully submerged(where the deck top or finished grade is at the surge water surface),as shown in Fig. 2. Water depth is fixed at the maximum surge con-dition in Hurricane Katrina.

Because the bridge deck is simply supported, a comparison be-tween uplift force and the weight of the bridge deck would providethe needed information to determine the cause of the deck dam-age. Also the percentage variations of wave forces at differentbridge elevations could be useful in assessing the potential riskof coastal bridges exposed to storm surge and wave attacks.

2. Storm surge and waves in Biloxi Bay during hurricane katrina

Tide and wave gauge data are not available in the vicinity of thebridge at Hurricane Katrina’s landfall. In order to obtain the reliableinformation on waves and water levels at the bridge site duringHurricane Katrina for numerical simulations of dynamic waveforces using the wave–bridge interaction model, large-scale, welltested numerical models have to be employed. By coupling the AD-vanced CIRCulation (ADCIRC) surge model and the Simulation ofWAves in Nearshore areas (SWAN) wave prediction model, Chenet al. [3] have hindcasted the storm surge and wind waves gener-ated by Hurricane Katrina on the northeastern Gulf of Mexico,including Biloxi Bay, Mississippi. They have found the combinationof strong winds, shallow water depth, and funneling effect of thecoastal geometry resulted in the record high surge elevations ofHurricane Katrina. In particular, the extremely shallow waterdepth of an ancient Mississippi River Delta played a significant rolein the extremely high surge in that region.

Chen et al. [3] set up the surge model using the bathymetry andtopographic data with 3-s resolution from the Coastal Relief DigitalElevation Model of the NOAA National Geophysical Data Center(NGDC). The dry-land boundary was extended from the 0 m con-tour to the 10 m contour above the Surge Sea Level (SSL) in orderto simulate the flooding. For the tide elevation boundary condition,the study used the records of the tide gauge at three different sta-tions near the offshore open boundary of the regional-scale model;

1392 H. Xiao et al. / Computers & Fluids 39 (2010) 1390–1400

for the wind input, the H*Wind dataset was adopted, which wasdeveloped by the NOAA’s Hurricane Research Division (HRD). Thecomputational domain for the regional-scale wave model was sim-ilar to the storm surge model that covers the area from Grand Isle,Louisiana to the Apalachicola Bay, Florida. The wind data used inthe ADCIRC model was also adopted for SWAN. The wave measure-ments at the two offshore buoys near the offshore open boundarywere used as the boundary conditions for SWAN. A nested domainof SWAN with the spatial resolution of 100 m was used to obtainthe wave field near the Biloxi Bay Bridge.

Results in Chen et al. [3] indicated that the maximum storm-surge elevation was 6.6 m above the North America Vertical Datum(NAVD), which exposed the bridge spans to the extreme wave im-pact. The significant wave height in the vicinity of the bridge was2.6 m, which resulted in strong horizontal and uplift wave loadson the bridge decks. The corresponding peak wave period wasfound to be 5.5 s. In addition, the hindcasted variation of waveheight across the bay near the bridge indicated that the waveswere the highest in the middle of the bay because of the largerwater depth. Model results of storm surge and waves from Chenet al. [3], who have carefully tested their coupled models againstmeasured high watermarks and wave data in deep water, were se-lected for the numerical simulations of dynamic wave loads on thebridge deck in this study. Table 1 lists the modeled surge and waveconditions near the bridge.

3. Wave-loading model

3.1. Governing equation

For wave propagation problems, water can be viewed as anincompressible viscous medium. In the wave and structure interac-tion process, the wave-induced fluid motion is typically turbulent,and the turbulence effect cannot be neglected in the wave-loadingmodel. In the case of turbulent flows with high Reynolds numbers,the required resolution for small-scale turbulent fluctuations is sohigh that a direct numerical simulation based on the Navier–Stokesequation is extremely difficult. Therefore, the RANS equations (i.e.Eqs. (1) and (2)) are used to describe the surge fluid motion and ak � e model (Eqs. (3) and (4)) is used as the turbulence closure forthe RANS equations.

RANS equations

@ui

@xi¼ 0 ð1Þ

@ui

@tþ uj

@ui

@xj¼ � 1

q@p@xiþ gi þ

@sij

@xjð2Þ

k � e model

@k@tþ uj

@k@xj¼ @

@xj

mt

rkþ m

� �@k@xj

� �þ G� e ð3Þ

@e@tþ uj

@e@xj¼ @

@xj

mt

reþ m

� �@e@xj

� �þ C1e

ek

G� C2ee2

kð4Þ

where

Table 1Storm surge elevation and wave parameters in the vicinity of the bridge.

Bay bottom elevation (NAVD) �0.86 mSurge elevation (NAVD) 6.59 mDeck bottom elevation (NAVD) 4.98 mSignificant wave height 2.60 mPeak wave period 5.50 s

sij ¼ 2 mþ Cdk2

e

!rij �

23

kdij rij ¼12

@ui

@xjþ @uj

@xi

� �mt ¼ Cd

k2

e

G ¼ 2mtrij@ui

@xj; Cd ¼ 0:09; C1e ¼ 1:44; C2e ¼ 1:92;

rk ¼ 1:0; re ¼ 1:3

in which k is the turbulent kinetic energy; e is the turbulent dissipa-tion rate; rij is the rate of strain tensor; dij is the Kronecker deltafunction; mt is eddy viscosity; ui is the velocity vector of the surgeflow; p is the pressure of the surge flow; gi is the ith componentof the gravitational acceleration; q is the fluid density.

3.1.1. Numerical discretationIn the wave-loading model, the computation domain is discret-

ized into regular rectangular cells. All scalar quantities, i.e., p, k, e,mt, and the VOF function F are defined at the center of the cells,and the vector quantities are defined at the center of the cell faces.The governing equations are discretized by finite difference meth-od (FDM).

The two-step projection method [4] has been used to solve theRANS equations. The first step is to solve an intermediate velocityunþ1

i ,

unþ1i � un

i

Dt¼ �un

j

@unj

@xjþ gi þ

@snij

@xjð5Þ

where the superscript indicates the time level and Dt is the timestep size. Eq. (5) is the forward time difference equation of the RANSequation without the pressure gradient term. The intermediatevelocity unþ1

i does not satisfy the continuity equation.The second step is to project the intermediate velocity field onto

a divergence free plane to obtain the final velocity, which satisfythe continuity equation:

unþ1i � unþ1

i

Dt¼ � 1

qn

@pnþ1

@xið6Þ

@unþ1i

@xi¼ 0 ð7Þ

Taking the divergence of (6) and applying (7) to the resultingequation yields Poisson Pressure Equation:

@

@xi

1qn

@pnþ1

@xi

� �¼ 1

Dt@unþ1

i

@xið8Þ

The second-order central difference scheme is applied to theviscosity term of the momentum equations, and a third-order up-wind difference scheme [6] was applied to convection term in or-der to reduce the effect of numerical viscosity. For example, in thex-direction, the convection term of the momentum u @u

@x is ex-pressed as follow:

Ux ¼uiþ1

2;j

dxi uLiþ1;j � uL

i�1;j

� � ; uniþ1;j > 0 ð9aÞ

Ux ¼uiþ1

2;j

dxiþ1 uRiþ1;j � uR

i�1;j

� � ; uniþ1;j < 0 ð9bÞ

where

uLiþ1;j ¼ un

iþ12;jþUL

1dui;j þUL2duiþ1;j ð10aÞ

uRiþ1;j ¼ un

iþ12;jþ 1�UR

2

� duiþ1;j �UR

1duiþ2;j ð10bÞ

duiþ1;j ¼ uniþ3

2;j� dun

iþ12;j

ð10cÞ

UL1 ¼

dxidxiþ1

ðdxi�1 þ dxi þ dxiþ1Þðdxi�1 þ dxiÞð10dÞ

H. Xiao et al. / Computers & Fluids 39 (2010) 1390–1400 1393

UL2 ¼

dxiðdxi�1 þ dxiÞðdxi�1 þ dxi þ dxiþ1Þðdxiþ1 þ dxiÞ

ð10eÞ

UR1 ¼

dxidxiþ1

ðdxi þ dxiþ1 þ dxiþ2Þðdxiþ1 þ dxiþ2Þð10fÞ

UR2 ¼

dxiþ1ðdxiþ1 þ dxiþ2Þðdxi þ dxiþ1 þ dxiþ2Þðdxiþ1 þ dxiÞ

ð10gÞ

For Poisson pressure Eq. (8), second-order center differencescheme is used, and the final discretized equation can be arrangedas

1Dxi

1qn

iþ12;j

pnþ1iþ1;j � pnþ1

i;j

Dxiþ12

!� 1

qni�1

2;j

pnþ1i;j � pnþ1

i�1;j

Dxi�12

!( )

þ 1Dyi

1qn

i;jþ12

pnþ1i;jþ1 � pnþ1

i;j

Dyjþ12

!� 1

qni;j�1

2

pnþ1i;j � pnþ1

i;j�1

Dyj�12

!( )

¼~uiþ1

2;j� ~ui�1

2;j

Dxiþ

~v i;jþ12� ~v i;j�1

2

Dyjð11Þ

As for stability criterion, a standard von Neumann’s analysis[14] is performed to obtain the following stability criteria:

Dt 6 minaDxUmax

;aDyVmax

�ð12aÞ

Dt 6 min1

2ðmt þ mÞðDxÞ2ðDyÞ2

ðDxÞ2 þ ðDyÞ2

" #( )ð12bÞ

The first constraint is set by the advection terms and the secondthe diffusion terms.

3.2. Boundary conditions

On the free surface, the dynamics free surface boundary condi-tion and the kinematic boundary condition must be satisfied. Forkinematic boundary condition, any particle cannot go though thefree surface. For dynamics free surface boundary condition, conti-nuity of stress components must be satisfied. The boundary condi-tions for the free surface assume the total derivative of anyphysical property associated with the free surface particles mustvanish on the free surface. In order to capture the wave surface,VOF method is adopted. On the wave-maker (wave-inlet) bound-ary, the wave surface displacement and the velocities are specifiedbased on the analytical solution of linear wave, g = gt, u = ut, v = vt,where ‘t’ is the given time (or phase).

The original VOF method for free surface boundary was pro-posed in the early 1970s. Reviews of VOF method can be foundin Rider and Kothe [18]. The donor–acceptor algorithm introducedby Hirt and Nichols [11] is the first. [23] algorithm uses a moreaccurate interface reconstruction method that Hirt and Nichols’VOF algorithm. Although Youngs’ idea first appeared in 1982, theoriginal paper had little detail of the methods by which the inter-face was reconstructed and fluxes calculated. Rudman [19] de-scribed the implementation of Young’ VOF technique in details.In this study, the Youngs’ VOF algorithm (1982) is used to updatethe VOF function and reconstruction the free surface during solu-tion during every time step.

Near wall function method [17] is adapted in this model alongthe rigid wall boundary. The universal logarithmic law of the wallwith smooth surface that is applicable to the fully turbulent regionoutside the viscous sub-layer is expressed as,

uus¼ 1

jln

usymþ C ð13Þ

where u is resultant velocity parallel to the wall at the first cell, us isthe resultant friction velocity, j = 0.41 is the constant, y is the nor-mal distance to the wall boundary, m is the kinetic viscosity of

water, and C is an empirical constant related to the thickness ofthe viscous sub-layer (C = 5.0 in a boundary layer over a smoothsurface; for rough walls, smaller values of C are desirable).

The standard logarithm wall function proposed by Launder andSpalding [17] has been widely used for many flow situations as de-scribed by Chen and Jaw [5]. For modeling turbulence in surfacewater flows (including waves), ASCE Task Committee on Turbu-lence Models in Hydraulic Computations [1] recommends the useof the logarithmic wall function as the semi-empirical approachfor practical application because it is sufficiently accurate for mostsurface water situations. Without using expensive fine grid resolu-tion, the logarithmic wall function effectively bridges the viscoussub-layer at the wall with the first numerical grid point placed out-side the sub-layer. It should be noted that the logarithm wall func-tion is valid only when the first grid point is placed within thelogarithmic region, i.e., when 30 6 y� ¼ usy

m 6 100. The grid systemfor all the computations presented in this study satisfies this crite-rion. Model validation to be described below confirms that thestandard logarithm wall function works well for coastal wave loadmodeling.

On the inlet wave-maker boundary, the wave surface displace-ment and the velocities are specified based on the analytical solu-tion, g = gt, u = ut, v = vt, where ‘t’ is the given time (or phase). Thevalues of k and e are specified on the wave-maker boundary inassumption of small disturbance. For the case study of HurricaneKatrina, information of horizontal velocity, vertical velocity, andfree surface displacement f were specified by analytical solutionsof linear wave theory. Wave parameters were determined from alarge-scale wave propagation model.

On the outlet open boundary without wave reflection, the Som-merfeld radiation condition is adopted, and a sponge layer [16] isset before the open boundary to absorb the wave energy. The Som-merfeld radiation condition allows the wave going out of the com-putational domain without significant reflection. The gradient ofall hydrodynamic variables are assumed to be zero at the rightboundary,

@Q@tþ Un

@Q@n¼ 0 ð14Þ

where Un is the phase celerity of wave at the open boundary and Qrepresents the hydrodynamic parameters such mean flow veloci-ties, free surface displacement, kinetic energy, dissipation rate.

3.2.1. Model validationTo support the numerical modeling study of wave uplift loading

on the Biloxi Bay Bridge, model validation was conducted by com-paring numerical results with experimental data. French [9] con-ducted extensive experiments to analyze the wave force oncoastal decks exerted by a solitary wave. In this section, thewave-loading model is used to compute the wave force of the sameproblem setup as French’s [9] experiments in order to test theaccuracy of the model in simulating uplifting wave forces.

Fig. 3 shows the experimental setup of French [9]. The stillwater depth is d = 0.381 m (15 ft); the width of rectangular struc-ture L = 4d (1.524 m); the distance from the bottom of the rectan-gular structure to the still water surface is s = 0.2d (0.0762 m). Theincident wave height is H = 0.24d (0.0914 m).

The computational domain covers the experimental setups. Thehorizontal length of the computational domain is 14 m, and thevertical length is 0.6 m. Fig. 3 also shows the coordinates of thecomputational domain. The coordinate of the left bottom point ofthe rectangular structure is (7.229, 0.4572) in the computationaldomain. Waves are generated in the left boundary, and propagatefrom the left to the right direction. The pressure and velocity arerecorded when the solitary wave crest reaches x = 3.5 m.

wL

Fig. 3. Experiment setup [9] for numerical model validation. (Fs is the weight of water in the approaching solitary wave above the platform, marked as the shaded water area).

1394 H. Xiao et al. / Computers & Fluids 39 (2010) 1390–1400

Model sensitivity studies have been conducted under differentgrid resolutions. Two different grid resolutions have been tested:Dx = 0.01 m, 0.02 m, and Dy = 0.005 m, 0.01 m. Results indicateno significant differences on wave forces. Therefore, the grid sizewith Dx = 0.02 m and Dy = 0.01 m is selected for validationsimulations.

Fig. 4 shows the comparison of computed and measured verticalcomponents of the wave force when the ratio of incident waveheight and water depth is 0.24. The total vertical hydrodynamicforce computed based on the present numerical model is normal-ized with respect to Fs. Fs is equivalent to the weight of water inthe region above the platform, denoted by the shaded water areain Fig. 3. It can be seen from Fig. 4 that the vertical wave force sim-ulated by the present model agrees with the experimental data ofFrench [9]. The model reasonably reproduced the negative forcesas the results of the Bernoulli effects, the phenomenon of internalpressure reduction with increased water velocity.

4. Modeling dynamic wave loads acting on Biloxi Bay bridgedeck

The wave-loading model is applied to investigate the waveloads on several typical decks of the Biloxi Bay Bridge during Hur-ricane Katrina. Bridge decks at three different elevations related tothe surge height were selected: Case I: emergent (where the deckor girder bottom is at the surge water surface), Case II: half sub-merged (where a half of the deck or girder height is below thesurge water surface), and Case III: fully submerged (where the decktop or finished grade is at the surge water surface), as shown inFig. 2. The typical wave and storm surge parameters at the bridgedeck location are obtained from the modeling results of Chen et al.[3] and summarized in Table 1.

Fig. 5 shows the detailed design of the bridge deck. The cross sec-tion of the bridge deck is simplified into a rectangular-shaped struc-ture in a two-dimensional computational domain, as shown in Fig. 6.The dimensions of the bridge deck are given below: w = 10.44 m is

0 2 4 6 8 10-4

-3

-2

-1

0

1

2

3Experiments (French, 1969)Present Method

sFF /

dgt //

Fig. 4. Comparisons of uplift force between experimental data and numericalmodel results. Ratio of wave height to depth (H/d) is 0.24. Fs is the weight of waterin the approaching solitary wave above the platform, marked as the shaded waterarea; d is the water depth.

the width of the span cross section; Hb = 1.35 m is the thickness ofthe bridge deck; d = 7.22 m (storm-surge elevation – seabedelevation) is the water depth under the storm surge condition.

The dimension of the computation domain is x = �250 m to250 m, y = 0–10 m. As shown in Fig. 6, the origin of the coordinateis fixed at the seabed. In the horizontal direction, the center of thebridge deck is placed at x = 0 m. Model sensitivity studies havebeen conducted under different grid resolutions. Three differentgrid resolutions have been chosen, Dx = 0.05 m, 0.1 m, 0.2 m, andDy = 0.025 m, 0.05 m, 0.1 m. Results indicate no significant differ-ences on wave profiles and forces. Therefore, the grid size withDx = 0.1 m and Dy = 0.05 m is selected for the remaining simula-tions. As shown in Fig. 7b, the smooth wave surface (before hittingthe deck) indicates that the model grid is fine enough to resolve thesurface wave profile. In all numerical simulations given in thismanuscript, waves are generated in the left boundary, and propa-gate from the left to the right direction.

A series of linear waves are generated at the left boundary usinganalytical results and propagate to the right. The fixed time step(dt = 0.1 s) is used to provide a stable iteration and precise solutionduring the entire computation. The velocity distribution and waveforce on the bridge deck are recorded from the moment the simu-lation starts.

4.1. Case I: storm-surge elevation at the bottom of the bridge deck

In this section, the selected bridge deck bottom is at the watersurface and the interaction between wave and bridge is simulatedby the wave-loading model. Fig. 7 shows the snapshots of velocitydistribution in the vicinity of the bridge deck within a wave period(5.5 s) during the wave–structure interaction. Fig. 7a presents thedeck position and still water depth when the wave is far away fromthe bridge. The deck bottom is just above the water surface, and awave crest is approaching the deck. In Fig. 7b, the wave crestcomes near the bridge, but still has no contact with the deck. Be-cause the bridge deck is located just above the water surface, thewave speed and profile are not affected until the wave front meetsthe deck in Fig. 7c. Fig. 7c also indicates the initiation of the inter-action between wave and bridge deck.

The wave crest comes nearer and hits the bridge deck front inFig. 7d, where the wave and deck interaction begins. Fig. 7d showsthe beginning of the interaction with flow separation occurring atthe lower front of the bridge deck. This interaction shows waterparticles at the wave crest with the maximum velocity. Meanwhile,as the flow speed is relatively high in front of the deck, advectiondominates local diffusion and an ellipse-shaped vortex is formed.Fig. 7e shows a snapshot of the wave crest reaching the middleof the bridge deck. At this time, water is climbing up the bridgeat the rear of the deck, and the flow pattern is quite similar to waverun-up on a vertical wall.

Fig. 7f shows the moment when the wave crest is leaving thebridge deck and another wave crest is approaching. While themajority of the first wave has completed the interaction with thebridge deck and approaches downstream, some of the wave energy

Fig. 5. Design of the US90 Biloxi Bridge Deck (westbound, in centimeter).

Fig. 6. Sketch of the model setup for simulating wave and bridge interaction in Biloxi Bay. (The coordinates are fixed at the seabed and the computation domain is defined as:x = �250 to 250 m, and y = 0–10 m.)

H. Xiao et al. / Computers & Fluids 39 (2010) 1390–1400 1395

is reflected by the bridge deck, shown by the large velocity magni-tude beneath the deck head (left side). Also, the water level at therear of deck is higher than (the water level) in the front as the wavecrest is passing the deck rear and the wave trough is coming to thedeck head. As a result, a back flow is formed along the deck bottomfrom the rear to the front. This reversed flow causes the vortex atthe deck front to gradually shed, and drift upstream. Fig. 7g showsthe wave profile when wave crest has just passed the bridge deck.A vortex pair also formed at the rear bottom of the deck. As theinteraction continues, this vortex pair sheds from the bottom ofthe deck, and drifts to the water surface.

It can be seen from Fig. 7 that the combined interaction amongthe approaching wave crest, the departing wave crest, and thebridge makes the flow pattern under the deck rather complex.

The time history of uplift and horizontal forces in several waveperiods (5.5 s) are shown in Fig. 8, with the solid, dash-dot, anddash-dot-dot lines representing the calculated uplift force, hori-zontal force, and bridge deck weight, respectively. It is seen fromthe figure that both the uplifting and horizontal forces occur at reg-ular intervals. This is because only the wave crest can exert waveforce on the bridge deck. When the wave trough comes, water sur-face falls below the initial equilibrium position, and makes no con-tribution to the wave force. Also, the time series of uplifting force ischaracterized by large positive force and very small negative force.The positive uplifting force is due to the advancing wave front,where the rising water makes initial contact with the bridge deckbottom. The phenomenon of negative pressure can be explained asfollows. As the trough advances towards the deck, the water fallsbelow the equilibrium position. This drop of water surface underthe deck bottom creates a vacuum in the space between the deckand the new water surface, and an equal volume of air fills this

space. This process creates a suction force as a negative uplift force.The modeling result shows the maximum uplift force for theemerged case (where the deck bottom is located at surge watersurface) is 78.3 metric tons, which does not exceed the bridge spanweight of 154 metric tons.

4.2. Case II: storm-surge elevation at the middle height of the bridgedeck

In this case, the selected bridge deck elevation is reduced by ½of the deck height (Hb = 1.35 m) compared with Case I, and thesurge water elevation is at the middle height of the bridge deck,as shown in Fig. 9a. The process of the interaction between waveand bridge deck at different phases within a wave period (5.5 s)is simulated and presented in Fig. 9. As the bridge is partially sub-merged under water, the presence of the bridge deck has a greaterinfluence on the propagation of the incident wave than the emer-gent case. It is evident in Fig. 9b, as the wave is approaching thebridge deck, the wave profile becomes asymmetric and the ampli-tude slightly increases because of the presence of the partially sub-merged bridge deck. The further steepening of the wave crestinitiates wave–bridge interaction, and the wave front begins tohit the front of the deck, as illustrated in Fig. 9c. The wave contin-ues to act on the deck as its turbulent crest moves toward the deck.Fig. 9d shows the moment when the wave crest is meeting thebridge deck. The deck reflects parts of the wave energy, and thereflection effect is stronger than that in the previous case (wherethe deck bottom is located at the surge water surface). Fig. 9fshows the completion of wave–bridge interaction, and the wavecrest moves just past the deck. In Fig. 9g, the wave elevation andflow speed are greatly reduced at the completion of the deck–

Fig. 7. Wave interactions with emerged bridge deck. (Case I: bridge deck bottom located at surge water elevation = emerged case.)

(b)

(c)

(d)

(e)(f)

(g)

(h)

Time (s)

Force(MetricTons)

4135.53025.4-100

-50

0

50

100

150

200

250

300 Uplifting forceHorizontal forceBridge span weight (154 Metric tons)

Fig. 8. Forces on emerged bridge deck (per span length). (Case I: bridge deckbottom located at surge water elevation = emerged case.) Points (b)–(h) corre-sponds to snapshots (b)–(h) of Fig. 7.

1396 H. Xiao et al. / Computers & Fluids 39 (2010) 1390–1400

water interaction. As a result, forces exerted on the deck alsodecrease.

Fig. 10 shows the time history of the horizontal and upliftingforces on the bridge deck in the process of the wave–bridge inter-action. In the horizontal direction, when the wave is relatively faraway, the bridge is only under the influence of hydrostatic pres-sure, and the total horizontal force equals zero. When the wave

front meets the bridge, the water level at the bridge front rises rap-idly. As a result, the total horizontal force increases gradually untilit reaches the positive maximum. After that, the horizontal forcedecreases gradually, changes its direction, and reaches its negativemaximum. In the vertical direction, the pattern of the uplift force isdifferent from the pattern in Case I (emerged). As the bridge deck ispartially submerged under water, the wave trough can also act onthe bridge deck. Therefore, the vertical force does not occur atintervals as shown in the previously emerged case. Instead, the up-lift force exhibits a harmonic pattern. It is noted from T = 30 toT = 31 in Fig. 10, which corresponds to Fig. 9b and Fig. 9c, the ver-tical force exerted on the bridge deck does not change much. This isbecause the wave elevation does not rise dramatically in this timeinterval, and the water particle speed is relatively small in thewave front. At T = 33, which corresponds to Fig. 9e, the verticalforce exerted on the bridge deck reaches its maximum. It is alsonoted that the vertical force does not exceed the weight of thebridge deck in this partially submerged case.

4.3. Case III: storm-surge elevation at the top of the bridge deck

In this case, the top of the selected deck is now in contact withthe surge water surface. Fig. 11 shows the process of interactionbetween the bridge deck and waves at different phases within a

Fig. 9. Wave interactions with partially submerged bridge deck. (Case II: bridge deck middle located at surge water elevation = partially submerged case.)

Time (s)

Force(MetricTons)

4135.53025.4-100

-50

0

50

100

150

200

250

300 Uplifting forceHorizontal forceBridge span weight (154 Metric tons)

(b) (c)

(d)

(e)(f)

(g)(h)

Fig. 10. Forces on partially submerged bridge deck (per span length). (Case II:Bridge deck middle located at surge water elevation = partially submerged case.)Points (b)–(h) corresponds to snapshots (b)–(h) of Fig. 9.

H. Xiao et al. / Computers & Fluids 39 (2010) 1390–1400 1397

wave period, where the bridge deck is completely submerged un-der the water surface. As the bridge deck is fully submerged, theeffect of wave–bridge interaction is more evident than that in pre-vious cases. It can be seen from Fig. 11b–d that in the initial stageof the wave–deck interaction, as the wave front has just reachedthe bridge deck, flow separation occurs at the bridge front and vor-texes are formed both at the bottom and top. As a large portion ofthe wave energy is distributed close to the water surface, the veloc-

ity of the water is relatively high at the upper layer of the water.Due to this strong advection in the vicinity of the bridge deck, vor-texes take on ellipse shapes. Afterwards, as the wave crest is leav-ing the deck and wave rough is coming to the deck front, the waterlevel at the rear rises while the water level at the front falls, asillustrated in Fig. 11f. This rise-fall pattern of water level causeswater to flow back to the front from the bridge rear. As a result,vortexes at the bridge front shed off and drift upstream.

Fig. 12 shows the time history of horizontal and uplift forces onthe bridge deck during the interaction between the wave and thebridge. For the horizontal wave force, when the wave front has justhit the bridge, the water level and velocity in front of the deck aremuch greater than those at the rear side. Consequently, the hori-zontal force increases during the first few seconds of the interac-tion. As the wave crest is traveling across the bridge, the waterlevel on the rear side of the deck increases gradually, and the hor-izontal force decreases. When the wave crest is leaving and thewave trough has just reached the bridge, the horizontal forcereaches the negative maximum. In the vertical direction, the initialvertical force on the bridge deck is the buoyancy force. As the wavetravels across the deck, the water level rises, and ultimately dropsto the start position. Consequently, the vertical force reaches itspositive and negative maxima. The weight of the bridge span is154 metric tons (340 kilo pounds), and the estimated maximumuplift force exerted on the bridge exceeds the weight of the bridgespan. Fig. 12 also suggests that the damage to the bridge deck was

Fig. 11. Wave interactions with fully submerged bridge deck. (Case III: bridge deck top located at surge water elevation = fully submerged case.)

Time (s)

Force(MetricTons)

4135.53025.4-100

-50

0

50

100

150

200

250

300 Uplifting forceHorizontal forceBridge span weight (154 Metric tons)

(g)(b)

(c)

(d) (e)

(f)

(h)

Fig. 12. Forces on fully submerged bridge deck (per span length). (Case III: bridgedeck top located at surge water elevation = fully submerged case.) Points (b)–(h)corresponds to snapshots (b)–(h) of Fig. 11.

1398 H. Xiao et al. / Computers & Fluids 39 (2010) 1390–1400

mainly caused by the wave uplift force as its magnitude from anindividual wave exceeded the weight of the simply-supportedbridge decks. The horizontal component of the wave force also con-tributed to the bridge failures because all bridge decks had nostrong connection to the substructure of the bridge to resist thehorizontal wave impact.

Chen et al. [3] have focused on the analysis of the failed bridgedecks that were not submerged by the flood water and concluded

that the uplift force generated by the most probably extreme wavewas greater than the weight of the bridge deck. Notice that themost probable extreme wave height is about 1.8 times the signifi-cant wave height. Nevertheless, their conclusion on the waveloads, rather than the buoyancy force, being responsible for thebridge deck collapse is consistent with the finding in this paper.

5. Effect of submersion depth on the maximum wave force

The maximum wave force exerted on bridge decks is dependenton the elevation of the bridge deck related to surge water eleva-tion. In this section, the effects of submersion depth on the maxi-mum wave force are examined by changing the deck elevationsubmerged under water and fixing all other variables (water depth,wave period, etc.). As shown in Fig. 2, the elevation view of thebridge is in arc shape. The changing of deck elevation also repre-sents different deck locations on the bridge under the same ex-treme wave and surge condition.

As shown in Fig. 13, the submersion depth (y) is defined as thedistance from the surge water surface to the bottom of girders sup-porting the bridge deck. The submersion coefficient (Cs) is definedas Cs = y/Hb, where Hb is the height of the bridge deck. Based on thedata summarized in Table 1, in the case of Hurricane Katrina, Cs = y/Hb = (6.36–4.98)/1.35 = 1.02 � 1.

In order to investigate the effect of submersion depth on themaximum uplift force, five different submersion depths in refer-

X

Y

o

w=10.44m

CASE IV

Cs = y/Hb =2

X

Y

o

w=10.44m

CASE III

Cs = y/Hb =1

y

d=7.22m

X

Y

o

w=10.44m

CASE I

Cs = y/Hb =0

Fig. 13. Model setups to investigate the effect of submersion depth. (Hb = 1.35 m is the height of the bridge span).

2.5

Weight of bridge span && (154 Metric tons)Maximum uplifting force

Case V

H. Xiao et al. / Computers & Fluids 39 (2010) 1390–1400 1399

ence to bridge height (Hb = 1.35 m) are considered, which areCs = y/Hb = 0.0, 0.5, 1.0, 1.5, and 2.0, respectively, as shown inFig. 13. The water depth (d) and significant height (Hs) remain asthe Hurricane Katrina condition, that is, d = 7.22 m andHs = 2.6 m. Other parameters, such as dimensions of the computa-tional domain, grid sizes, and time steps remain unchanged. Foreach of the submersion depths, the process of interaction betweenthe wave and bridge deck is simulated, and the time history ofwave force exerted on the bridge deck in a wave cycle is obtained.

The time history of the total uplift force per span length exertedon the bridge is shown in Fig. 14 for various submersion depths:Cs = y/Hb = 0.0, 0.5, 1.0, 1.5, and 2.0. The total force per span lengthis defined as the sum of the products of the computed uplift pres-sure, and the contact area of the deck span.

In Fig. 14, the total uplift force computed based on the presentnumerical model is normalized with respect to Fb (Fb = 154 metrictons is the weight of a bridge deck), and the time is normalized byffiffiffiffiffiffiffiffi

d=gp

(d = 7.22 m is the depth of water in the storm surge associ-ated with Hurricane Katrina). For all the five submersion depths,the normalized total uplift force in a wave cycle increases withtime to a maximum, then decreases with time until it reachesthe minimum point. It is noted in Fig. 14, when the wave crestfront is hitting the bridge deck, the total uplifting wave force ex-erted on the deck starts to go up; when the wave completes itsinteraction with the deck and is leaving the bridge, the upliftingwave force reaches its maximum; when the wave crest has left

45 50 55 60-1

-0.5

0

0.5

1

1.5

2 Cs=0.0Cs=0.5Cs=1.0Cs=1.5Cs=2.0BridgeWeight (154 Metric tons)

gdT /

b

uplift

F

F

Fig. 14. Uplifting wave force (per span length) exerted on bridge deck underdifferent submersion depth. (Uplifting force Fuplift is normalized by Fb; Fb = 154metric tons is the weight of the bridge deck; time T is normalized

ffiffiffiffiffiffiffiffid=g

p; d = 7.22 m

is depth of water during Hurricane Katrina; Cs = y/Hb is the coefficient ofsubmersion depth.)

and wave trough is reaching the bridge front, the uplift wave forcedrops to the minimum.

To better understand the effect of submersion depth, the max-imum positive uplift force for each of the time history is obtainedand plotted in Fig. 15. By comparing the results shown in Fig. 15,one can see the effect of submersion depth on the maximum upliftforce. The curve indicates the maximum positive normalized totaluplift forces per span width are strongly dependent on the submer-sion depth. As the submersion depth increases, the maximum nor-malized positive force increases to a peak value then dropsgradually. As shown in Fig. 15, the maximum uplifting wave force,when the bridge deck or girder bottom is at surge water elevation(Cs = 0.0), is 0.51 of the bridge span weight. From Cs = 0.0, the max-imum uplift force starts to increase. When the submersion coeffi-cient increases from Cs = 0.5 to 1.0, the normalized maximumuplift force increases from 0.85 to 1.36. At about Cs = 1.1, the max-imum uplifting force reaches its maximum (which amounts for287% of the initially emergent case), and then starts to drop withsubmersion depth. As is evident from Fig. 15, with the increase

Submersioncoefficient(Cs)

0 0.5 1 1.5 2-0.5

0

0.5

1

1.5

2

Case I

Case II

Case III

Case IV

Normalized uplifting forcebFF /max

Fig. 15. Variation of maximum uplifting force with submersion coefficient. Themaximum uplifting force Fmax is normalized by the weight of bridge span Fb = 154metric tons. Cs = y/Hb is the coefficient of submersion depth.

1400 H. Xiao et al. / Computers & Fluids 39 (2010) 1390–1400

of submersion coefficient from Cs = 1.5 to 2.0, the maximum uplift-ing force drops from 1.31 to 0.91 of the bridge span weight.

This phenomenon can be explained as follows. When the bridgedeck bottom is level with the surge water surface, i.e. Case I withCs = 0.0, only the wave crest has direct contact with the bridgedeck, and a large portion of the wave energy cannot be conveyedto the deck. Therefore, the maximum uplift force is comparativelysmall at Cs = 0.0. When the submersion coefficient Cs increases toaround 1.0, a majority of the wave force is concentrated in thevicinity of the water surface, and the maximum uplift force occursat this submersion depth (when the bridge deck is just fully sub-merged). Further increase in the submersion depth causes a de-crease in the wave energy impacting the deck. Moreover, thewave overtopping increases the weight of water on top of thebridge deck. As a result, approximately from Cs = 1.0, the maximumuplift force decreases with the submersion depth.

6. Conclusion

A wave-loading model based on the unsteady RANS equationswith a turbulent closure has been employed to simulate the inter-action between wave and bridge deck. Satisfactory comparisonswith experimental data of wave forces on horizontal platformshave shown that the model is capable of simulating the time seriesof wave loads acting on the bridge deck for bridge safety analysisand damage investigations. For the case study of wave loads onthe bridge deck of the Biloxi Bay Bridge during Hurricane Katrina(2005), numerical simulations indicate that uplift force was greaterthan the weight of the bridge deck for about 20–30% of the timeduring a wave period. As a result, the simply-supported bridgedeck was lifted and moved away by the combined vertical and hor-izontal wave forces.

In order to analyze the effect of submersion depth on the upliftforce, the interactions between a linear wave train and the bridgedeck with five different elevations (Cs = y/Hb = 0.0, 0.5, 1.0, 1.5, and2.0) have been studied using the present wave-loading model. Thetime histories of wave force on the bridge deck have been obtainedand analyzed. The results show that the maximum uplift force isthe largest when the bridge deck is fully submerged (Cs = 1.0), orthe surge water elevation is at the deck top. The maximum upliftforce reached 137% of the bridge span weight (154 metric tons).From this bridge elevation Cs = 1.0, with the increase of submersiondepth from Cs = 1.5 to 2.0, the maximum uplifting force is reducedfrom 131% to 91% of the span weight; and with the decrease of sub-mersion depth from Cs = 0.5 to 0, the maximum uplift force is alsoreduced from 84% to 50% of the span weight. Although the presentwave-loading model is a single-phase two-dimensional model, itimproves our understanding of wave forces at varying bridge ele-vations and wave–bridge interaction, which is very useful for theassessment of the potential risk of coastal bridges exposed tostorm surges and extreme wave conditions. The model present in

this study can also provide a quicker assessment of wave forces be-fore conducting further more expensive 3D numerical modeling orlarge-scale physical experimental studies.

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