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Dynamic analysis of an inflatable dam subjected to a floodK. Lowery, S. Liapis

Abstract A dynamic simulation of the response of anin¯atable dam subjected to a ¯ood is carried out to de-termine the survivability envelope of the dam where it canoperate without rupture, or over¯ow. The free-surface ¯owproblem is solved in two dimensions using a fully non-linear mixed Eulerian-Lagrangian formulation. The dam ismodeled as an elastic shell in¯ated with air and simplysupported from two points. The ®nite element method isemployed to determine the dynamic response of thestructure using ABAQUS with a shell element. The prob-lem is solved in the time domain which allows the pre-diction of a number of transient phenomena such as thegeneration of upstream advancing waves, the dynamicstructural response and structural failure. Failure takesplace when the dam either ruptures or over¯ows. Stressesin the dam material were monitored to determine whenrupture occurs. An iterative study was performed to ®ndthe serviceability envelope of the dam in terms of the in-ternal pressure and the ¯ood Froude number for two ¯ooddepths. It was found that existing in¯atable dams are quiteeffective in suppressing ¯oods for a relatively wide rangeof ¯ood velocities.

Symbolsh, d The initial upstream water height

(depth)L Tank length for the ¯uid solutionFn � UB=

����������g � h

pFroude number (non-dimensionalvelocity)

UB Upstream velocity (towing velocity)R Semi-circular obstruction height

(radius)g The acceleration due to gravityq Density of water

a � Rh Non-dimensional dam or obstruction

heightt Time/ The velocity potentialw The stream functionr The gradient operatorD/Dt The substantial derivativex, y The x and y Cartesian coordinates of a

pointn The unit normal to a surfacen1 The x component of the unit normal to

a surfacen2 The y component of the unit normal to

a surfacei The imaginary number given by

p�ÿ1�or an integer depending on context

z � x� iy The complex coordinateb � /� iw The complex potentialC In¯uence coef®cientLd Desingularization distancep The value for Pi (3.141592654...)c The local wave velocityP Pressureg Free surface elevationh Angle varying from zero on the leading

edge to p on the trailing edge of theobstruction

Cp � P=qgh Pressure coef®cientCd � drag=qgh2 Drag coef®cientC1 � lift=qgh2 Lift coef®cientPE Potential energyKE Kinetic energy

1IntroductionIn¯atable dams are cylindrical in shape and attached to afoundation strip. They can be in¯ated with air, ®lled withwater, or pressurized with a combination of air and wa-ter. Their height can be up to six meters and their lengthup to one hundred ®fty meters. They may be anchored toa speci®cally made reinforced-concrete foundation stripor an existing base, like the crest of a dam. They areusually constructed of a nylon reinforced polymer. In-¯atable dams are relatively easy to install, do not corrode,require little maintenance, and can handle extreme tem-peratures. They can be covered with impact resistant tilesto prevent them from being punctured by gun®re andother debris.

Computational Mechanics 24 (1999) 52±64 Ó Springer-Verlag 1999

52

K. LoweryAker Engineering, 11757 Katy Freeway,Suite 1300, Houston, TX 77079

S. Liapis (&)Offshore Structures, Shell Oil Company,3737 Bellaire Blvd, Houston, TX 77001

Most of this research was carried out while both authors wereat Virginia Polytechnic Institute and State University. It waspartially supported by a grant from the National ScienceFoundation (grant no. CMS-9422248). Thanks are due toProfessor R.H. Plaut for many helpful comments.

In¯atable dams are currently used in a number of watercontrol applications. These ¯exible structures may be usedto temporarily store water or serve as dykes. In permanentapplications, they can be used to direct water in irrigation,control water level in a hydroelectric facility, or raise theheight of an existing dam. Several thousand in¯atabledams are in use today. A 640-meter long string of sevenin¯atable dams is in place on the Susquehanna River inPennsylvania to create Lake Augusta. Each segment isconnected by concrete pylons to create a continuous dam.Some of the ¯ow in the Santa Ana River in California isdiverted by an in¯atable dam to recharge groundwater. Alarge project to create a 220 surface-acre (850 feet wide by2 miles long) lake in Tempe Arizona using in¯atable damsis currently underway. Figure 1 shows an artist's renditionof this project. The city will construct a string of many 240feet long in¯atable dam sections ranging from six to six-teen feet in diameter. A one-inch wall thickness will beused over the entire dam. It is interesting to note that onesection of the dam will be allowed to over¯ow at a con-trolled rate to produce a scenic cascade. Water will then bepumped back into to the lake to prevent runoff. The totalconstruction cost of the dam is estimated to be fortymillion dollars. Many other examples of successful damscan be found worldwide. A few in¯atable dams have failedduring service. The Mangla Dam in Pakistan (see Binnieet al. 1973), became dynamically unstable and failed underover¯ow conditions.

Currently, in¯atable dams are not used to suppress¯oods in emergencies. It is desirable to extend their use forthis purpose. These dams could be placed along riverbanks to protect homes, businesses, industries, and townsfrom ¯ooding. They could be de¯ated when not needed, toprovide access and views, and then in¯ated when ¯oodingis imminent. This requires some research to predict thedynamic response of the structure to ¯ood conditionswhich is the objective of the present study.

Several authors have reported analytical results on in-¯atable dams. For example, Hsieh and Plaut (1990), andAl-Brahim (1994) analyzed the free vibrations of in¯atabledams in two dimensions while Dakshina-Moorthy et al.(1995) and Mysore et al. (1998) studied free, three-dimensional vibrations of in¯atable dams. Alwan (1986,1988) performed modal analysis of an in¯atable damsubjected to a ¯ood. Wu and Plaut (1996) analyzed smallvibrations of an in¯atable dam under over¯ow conditions.

Also related to the present work, are studies of thedynamic characteristics of structures in air or in contactwith water which have been carried out for a number ofdifferent practical applications. Kjellgren and Hyvarinen(1998) used a ®nite element numerical solution of theNavier-Stokes equations to compute the ¯ow aroundmoving structures such as sports car sections in twodimensions. Experimental data as well as numerical pre-dictions of the natural frequencies and mode shapes of a¯exible cylinder vibrating in air and in water were pre-sented by Ergin et al. (1992). The dynamic response of aoffshore platform subjected to random forces from waveswas analyzed by Venkataramana and Kawano (1996). In arecent study, Nawrotzki et al. (1998) analyzed the elasticand inelastic response of shells using incremental path-following algorithms.

In this study, a dynamic simulation of the response ofan in¯atable dam subjected to a ¯ood is carried out. Un-like the previous studies which solved the problem in thefrequency domain, in the present work a fully nonlineartime-domain analysis is used. This allows for the predic-tion of large unsteady motions of the dam including caseswhere the dam ruptures or over¯ow occurs.

The problem of a ¯ood impinging on an in¯atable damis assumed to be two-dimensional. In this study, a ¯ood istaken to be a uniform stream of water striking the dam.The ¯ood is assumed to start with a uniform depth of 2.0or 3.0 meters, and a uniform velocity. The ¯ow a suf®cientdistance ahead of the dam retains this initial velocity anddepth. The analysis begins when the front of the ¯ood iscompletely in contact with the dam.

The dam is anchored to the bottom at two points usingpinned boundary conditions. Its stress free shape (initialshape) is that of a large semicircle. The dam has a uniformthickness and is comprised of nylon reinforced rubber.The dimensions and material properties of the structureused in the present study are representative of existingstructures. Figure 2 illustrates the design of a typical in-¯atable dam. The dam is supported by an applied internalpressure ranging from a pressure of 1.0 to 5.0 meters ofwater. This internal pressure provides most of the stiffnessrequired to repel a uniform ¯ood applied to one side of thedam.

This problem has two distinct parts: nonlinear free-surface ¯ow, and elastic structural dynamics. These twoproblems are coupled and must be solved simultaneously.Transient, fully nonlinear computations for the ¯uid ¯oware performed using a mixed Eulerian-Lagrangian for-mulation. The dam is modeled as an elastic shell in¯atedwith air and simply supported from two points. The ®niteelement method is employed to determine the dynamicresponse of the structure using ABAQUS with a shell

Fig. 1. Artist's rendition of the in¯atable dam built to create a220 surface-acre man made lake in Tempe, Arizona

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element. A numerical towing tank with the in¯atable damon the downstream boundary and open boundary condi-tions on the upstream vertical boundary has been devel-oped to simulate the ¯uid ¯ow. The computer simulation``tows'' the ¯exible dam at a steady Froude number. The¯uid dynamics program determines the free-surface ele-vation and calculates the external pressure acting on thedam surface at each time step. This pressure is then usedas an input by the ABAQUS model at each time step todetermine the deformation and the velocities along thedam surface and the stresses in the dam. This informationis used by the ¯uid dynamics program in the next timestep to continue the dynamic analysis. The data for theresponse of the free surface is translated to a coordinateframe moving with the structure to show how the freesurface develops over time.

The simulation was run over a range of Froude numbersto determine the internal pressures where the structurefailed. Failure takes place when the dam either ruptures orover¯ows. Rupture is caused by too high an internalpressure. Over¯ow, on the other hand, is caused by too lowan internal pressure, or too high a Froude number. Thus, aserviceability envelope is formed where the dam can op-erate safely. Two envelopes were developed correspondingto upstream ¯ood depths of 2.0 and 3.0 meters.

The paper is organized as follows. Section 2 contains themathematical formulation and numerical procedures foranalyzing the ¯uid ¯ow and the structural response. InSect. 3 the numerical results for the ¯uid ¯ow, the struc-tural response and the operational envelope are presented.Concluding remarks are given in Sect. 4.

2Fluid±structure interaction studyThis problem can be separated into two parts at each timestep. The ¯uid dynamic ¯ow will be described ®rst, fol-lowed by the structural analysis. The last section willdescribe how the solutions of these problems are coupledto produce the fully dynamic simulation.

2.1Mathematical formulation of the fluid flowThe unsteady, two-dimensional potential ¯ow of an in-viscid, incompressible ¯uid induced by the motions of the¯exible boundary, is analyzed. Under the usual assump-tions of a two-dimensional, inviscid, irrotational ¯ow, the

problem may be described by a velocity potential, /, and astream function, w. The velocity potential, /, is de®ned sothat the ¯uid velocity vector is given by:

~V � ~r/

Instead of analyzing a uniform ¯ow incident on the ¯exiblestructure, the interaction of an initially stationary ¯uidwith a ¯exible structure moving to the left at a constantspeed, UB is considered. A stationary x-y coordinate sys-tem is used with y positive upwards and the x axis coin-ciding with the upper left corner of the undisturbed freesurface.

The ¯uid domain is bounded by the free surface, the¯exible boundary surface, the bottom and the far-®eldupstream boundary. Figure 3 shows the numerical tanklayout. Non-dimensional variables are chosen such thatthe initial ¯uid depth, the acceleration of gravity and the¯uid density are equal to one. The ¯uid ¯ow solution isnon-dimensionalized to allow easy application of variousdepths. The ¯uid ¯ow problem may be described in termsof two non-dimensional parameters. A depth-based Fro-ude number is de®ned as Fn � UB=

����������g � h

p, where UB is

the uniform upstream velocity, h is the uniform upstreamdepth, and g is the gravitational acceleration. The non-dimensional dam height is given as a � R

h where R is theradius of the dam.

In the ¯uid domain R, the velocity potential /, mustsatisfy the Laplace equation:

r2/ � 0 �1�On the free surface, CF, the kinematic boundary conditionsare:

DX

Dt� o/

oxand

DY

Dt� o/

oy�2�

where X, Y are the x and y coordinates of the particle onthe free surface. The dynamic free-surface condition isgiven by Bernoulli's equation:

D/Dt� ÿg� �r/�2

2�3�

where D is the substantial derivative, x and y are the co-ordinates of the position of a ¯uid particle, and g is the

↑↑

↑↑

↑↑

↑

↑ ↑ ↑ ↑ ↑

↑↑

↑↑

↑↑↑↑↑↑↑↑↑↑

↑↑

↑↑

↑↑

↑

↑ ↑ ↑ ↑ ↑

↑↑

↑↑

↑↑↑↑↑↑↑↑↑↑Internal Pressure

ThicknessMaterial properties:Modulus of elasticityPoisson's ratioCritical (failure) stress

Flood velocityDep

th

Fig. 2. Sketch of a typical in¯atable dam illustrating some basicparameters in¯uencing dam performance

Upstream boundarywith open boundaryconditions (CU)

Right hand boundary is translatedto the left at the chosen floodvelocity (V) and deforms with dam.

The fluid boundary is descretized into straightline segments or panels joined together at nodes.Note: only some nodes are shown for illustration.

Free surface (CF)

V

Fig. 3. Numerical towing tank layout

54

free surface elevation. On the ¯exible boundary surface,CS, the ¯uid ¯ow satis®es the no-penetration condition:

o/on� ow

os� �ud ÿ UB�n̂1 � vdn̂2 �4�

where UB is the velocity of the boundary, s is the arclengthalong the boundary and n is the unit normal with com-ponents n1 and n2 in the x and y directions respectively. ud

and vd are the components of the velocity of the ¯exiblestructure due to deformation in the x and y directions,respectively. On the ¯at bottom, CB:

o/on� ow

os� 0 �5�

The complex coordinate is de®ned as z � x� iy, and thecomplex velocity potential, b, as b�z� � /�z� � iw�z�.Since b is analytic in the ¯uid domain, it satis®es Cauchy'sintegral theorem:I

C

b�z��zÿ zk� dz

�0 for zk outside the fluid domain �6a�ÿiak b�zk� for zk on the fluid boundary �6b�2pib�zk� for zk inside the fluid domain �6c�

8><>:where the contour C consists of the free surface, CF, thebottom, CB, the ¯exible boundary surface, CS, and theupstream boundary. zk is an arbitrary control point. Thecontour C is traversed such that the region R always lies toits left and ak is the angle between two elements adjacentto zk on the contour. For a smooth contour ak � ÿp.

A mixed Eulerian-Lagrangian (MEL) formulation isemployed to solve the boundary value problem (1)±(5).This method has been used very successfully to predict avariety of nonlinear wave phenomena by a number ofinvestigators (Longuet-Higgins and Cokelet 1976, Vinjeand Brevig 1981, Lin 1984, Cointe 1989, Muthedath 1992).This method involves an Eulerian and a Lagrangian step.At each instant of time, the integral equation (6b) is solvednumerically (Eulerian step). On some portions of theboundary u is known (C/ boundary) while on others w isknown (Cw boundary). On the free surface u is given byEq. (3) while on the ¯at bottom w is zero and on thebottom obstacle surface w is given from Eq. (4). Aftersolving the boundary value problem at each time step, thefree surface conditions (2) and (3) are stepped forward intime to update the positions and the values of the velocitypotential of particles of ®xed identity (Lagrangian step).Only a brief description of the MEL formulation will begiven here. More details may be found in Vinje and Brevig1981 and Muthedath 1992.

2.1.1Numerical implementationThe boundary C is divided into elements (panels). Figure 2shows the layout of the tank and boundary elements.Within each panel, the complex potential, b�z�, is assumedto vary linearly with z. This can be expressed as:

b�z� � zÿ zj

zjÿ1 ÿ zjbjÿ1 �

zÿ zjÿ1

zj ÿ zjÿ1bj �7�

This approximation of b�z� reduces the problem of ®ndinga continuous potential to determining the unknown nodalvalues bj. These are found by collocation, that is, by sat-isfying the integral Eq. (6) at the panel vertices. The resultis a set of linear equations that can be solved for the un-known bs

j . If Eq. (7) is substituted into Eq. (6a), a dis-cretized form of the integral equation is found:I

C

b�z��zÿ zk� dz �

XN

j�1

Ck;jbj � 0 �8�

where the in¯uence coef®cients are given by:

Ck;j � zk ÿ zjÿ1

zj ÿ zjÿ1

� �ln

zj ÿ zk

zjÿ1 ÿ zk

� �

� zk ÿ zj�1

zj ÿ zj�1

� �ln

zj�1 ÿ zk

zj ÿ zk

� ��9�

zj and zk are the nodal and control point locations, re-spectively.

The ¯uid boundary consists of portions where / isknown (C/ boundary) and portions where w is known (Cwboundary). At the free surface, the velocity potential / isknown (C/ boundary) while on the bottom, the upstreamboundary, and the ¯exible surface of the structure thestream function w is known (Cw boundary). The applica-tion of the boundary conditions is done by setting theappropriate terms in Eq. (8) to zero as follows:

ReXN

j�1

Ck;jbj

( )� 0 for zk on C/ (10a)

Re iXN

j�1

Ck;jbj

( )� 0 for zk on Cw (10b)

Special attention to the nodes at the intersection of the freesurface with the upstream boundary and the surface of thestructure is required. At these points, / and w are com-pletely known. No equation corresponding to / or w onthat point is used. This has been shown to be an effectivetreatment (see Lin 1984).

The ¯exible obstruction making up the right handboundary is translated towards the left end of the tank tosimulate ¯ood conditions. Time stepping of Eqs. (2) and(3) is performed by Hamming's fourth order predictor/corrector method. A Runge-Kutta fourth order method isemployed for the ®rst four time steps to provide the initialstarting data for use with Hamming's method. A non-dimensional time step of 0.05 seconds is used. The struc-ture is started from rest with a velocity ramp. During thisacceleration phase, the velocity is gradually increased in alinear fashion to its ®nal value. It was found that a velocityincrease of 0.05 Fn per time step provides a smoothstartup.

Detailed accuracy and convergence tests were carriedout to determine the numerical tank length, the panelarrangement and the time step size for the numericalsimulations. Since the right boundary is moving to theleft, the length, L of the numerical towing tank reduceswith time. For the numerical results presented in this

55

study, the length of the tank was chosen equal to 40times the ¯uid depth. This value of L was suf®cient toprovide the desired results before the computationaldomain became too short. A total of 509 nodes were usedfor the simulations presented here. 300 nodes were usedon the free surface, 6 were used on the upstream boun-dary, 201 were used on the bottom boundary and 6 wereused on the surface of the structure. This created 299, 5,200, and 5 panels on the free surface, upstream boun-dary, bottom boundary, and structure surface, respec-tively. The nodes were spaced to provide uniform panellengths on the computational boundary. Note that thepanel lengths are not necessarily equal on the boundarysegments.

2.1.2Smoothing and node distribution on the free surfaceNodes on the free surface and the bottom boundary wereredistributed at each time step to retain equal panellengths on these boundaries as the tank length decreased.

Nodes on the free surface are smoothed every ®ve timesteps using a ®fth order smoothing formula developed byLonguet-Higgins and Cokelet (1976)

�yi � 116 ÿyiÿ2 � 4yiÿ1 � 10yi � 4yi�1 ÿ yi�2� � �11�

where the subscript i indicates the ith node, iÿ 1 is theprevious node and so forth. Smoothing is necessary tosuppress short wave, saw-tooth instabilities that are re-lated to numerical error. This formula worked very welland suppressed all saw-tooth instability seen in this study.No adverse effects of smoothing such as introduction ofnumerical damping or dissipation to the numerical solu-tion were observed.

2.1.3Upstream boundary conditionA satisfactory treatment of the upstream boundary is offundamental importance to the success of the numericalsimulations. In this work, a Sommerfeld radiation condi-tion is imposed at the upstream boundary in a mannersimilar to what has been used in other problems (Orlanski1976). The following condition is imposed:

owot� c

owox� 0 �12�

where c is the local wave velocity. This condition makesthe upstream boundary transparent to all waves of ve-locity c, thus eliminating re¯ections. The local wave ve-locity, c is in the general case unknown. Some authors(Han et al. 1983) suggested a numerical method of cal-culating c. In the present study, c is set equal to the phasevelocity

�����gh

p � 1 (unit depth and gravitational accelera-tion have been assumed). The numerical method used toapply Eq. (12) involves adding a column of points justinside the upstream and downstream boundaries. Then a®nite difference scheme is employed to ®nd the spatialand temporal derivatives of w�z�. For the time derivative,a backward difference scheme was used. For the spatialderivative, a forward difference scheme was used. Thisgives:

w�xu; yu; t� ÿ w�xu; yu; t ÿ Dt�Dt

� cw�xu � Dx; yu; t ÿ Dt� ÿ w�xu; yu; t ÿ Dt�

Dx� 0

�13�Rearranging to determine the nodal value of w�xu; yu; t�,w�xu; yu; t� � w�xu; yu; t ÿ Dt�

ÿ cDt

Dxw�xu � Dx; yu; t ÿ Dt��

ÿ w�xu; yu; t ÿ Dt�� �14�where xu, yu is the coordinate of a node on the upstreamboundary (xu � 0), t is the current time in the simulation,Dt is the time step interval, and Dx is the distance of theadditional points from the upstream boundary. The valueof Dx used was 5 times the upstream depth. As usual, thisoptimum value for Dx was determined by trial and error.This scheme was tested, using both ®rst and second orderdifference schemes to approximate the derivatives of w�z�,and various ways to determine the propagation velocity, c.The method shown and the simple choice of c � 1 wasfound to be superior and reduces the re¯ected waves to aminimum.

At each time step (t ÿ Dt) after solving for the value ofw on the boundary nodes, the value of w on the interiorpoints, w�xu � Dx; yu; t ÿ Dt� is determined using Eq. (6c).Then Eq. (14) is used to update the value of w�xu; yu; t� atthe nodes making up the upstream boundary at each timestep.

2.1.4Calculation of the pressure acting on the damThe pressure over the surface of the structure is foundusing Bernoulli's equation:

P ��ÿ ou

otÿ �r/�2

2ÿ �yÿ g�

��15�

In order to evaluate the time derivative of the velocitypotential, ut, we observe that the time derivative of thecomplex velocity potential, bt, is analytic in the ¯uid do-main. Therefore at each time step in a manner similar tothe b problem, Eq. (6b) is solved for the unknown valuesof bt. When solving for the time derivative bt, on some ofthe boundaries ut is known while on others wt is known ,similar to the b problem. On the free surface ut is given byEq. (3) as:

o/ot� ÿgÿ �r/�2

2

On the far-®eld boundaries wt is given from Eq. (12). Onthe ¯at bottom wt is zero, while on the structure a divideddifference formula is used to evaluate wt .

2.2Structural analysisThe dam is modeled as a thin elastic shell and is analyzedusing the ®nite element method. The ®nite elementpackage ABAQUS is used for both the static and dynamic

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analysis. One hundred four-node thin shell elements (theS4R type described in the ABAQUS manual see Hibbittet al. 1994) arranged along the circumference are em-ployed. The elements have a uniform thickness of 1.27 cmand uniform material properties. The dam is anchored tothe bottom at two points using pinned boundary condi-tions. Its stress free shape (initial shape) is that of a largesemicircle. It is comprised of nylon reinforced rubber andhas a uniform thickness of 1.27 cm. The dam is supportedby an applied internal pressure ranging from a pressure of1.0 to 5.0 meters of water. This internal pressure providesnearly all the stiffness required to repel a uniform ¯oodapplied to the one side of the dam. Table 1 contains thedimensions and material properties of the dam used in thisstudy. These values are typical of existing structures.

The end points of the structure are anchored to thebottom using pinned boundary conditions. Since theproblem is two-dimensional, the nodes are restrained frommoving in the z direction. A uniform pressure is applied tothe inside face of each element. The compressibility of theinternal air was taken into account by assuming that itbehaves isothermally. Hence, the internal pressure is notconstant but is determined from the relation (pressure ´volume) = c. The isothermal constant c is determined bytaking the product of the initial prescribed pressure andthe static volume of the dam. Then the value for the in-ternal pressure at any point during the simulation may befound by dividing this constant by the new volume of thedam. The pressure caused by the effects of the external¯uid (¯ood) is subtracted from the internal base pressureto obtain the pressure exerted on the dam. This pressure isdistributed appropriately over elements making up thedam in the ABAQUS model.

2.3Fluid±structure interactionValues for the structural properties of the dam material,initial dam shape, the initial (base) internal dam pressure,the ¯ood Froude number and the initial ¯ood depth areinput into a preprocessor. The preprocessor takes thesevalues and creates the ABAQUS ®nite element model.Since the dam is initially started from rest, a static analysismust be performed ®rst to determine the static shape ofthe dam produced by the in¯uence of the external hy-drostatic pressure and the internal in¯ation pressure. Thehydrostatic pressure is found directly from the formula:Pstat � qgdave, where q is the density of water (1000 Kg/m3), g is the gravitational acceleration (9.81 m/s2), and dave

is the average depth of each submerged element in theABAQUS model. Figure 4 illustrates the application of thepressures on the structural model. The total pressureacting on each element is found by subtracting the staticpressure, Pstat, from the internal pressure, Pint.

Once the static shape of the dam is found, the dynamicsimulation is started. The translational velocity of the damis smoothly started from rest to the ®nal Froude numberdesired for each run. The two parts of the problem, thenonlinear free-surface ¯ow and the structural deformationare coupled and must be solved simultaneously. The ¯uid¯ow depends on the structural deformation while thestructural deformation depends on the ¯uid forces. Thereare many different methods for simulating such coupled¯uid-structure interaction problems. In the present work,the ¯uid ¯ow problem is solved ®rst at each instant of timet. The new free surface shape, and the new pressure dis-tribution acting on the dam are obtained. The total pres-sure acting on each element of the structure is found bysubtracting the external pressure, Pdyn found by the ¯uidscode from the isothermal internal pressure, Piso:Ptot =Piso ) Pdyn. The total pressure is used to create an ABA-QUS restart ®le to perform a dynamic analysis over onetime step. The ABAQUS dynamic analysis computes thestresses in the dam material as well as the new dam shape,and the velocity distribution for the next time step(t � Dt). After updating the shape of the structure, the newvalues of the stream function w are found by integratingEq. (4) numerically. At the next time step (t � Dt) theentire procedure is repeated.

Figure 4 illustrates this method of solving the coupled¯uid-structure interaction problem. More involvedschemes to couple the ¯uid and structural solutions wereconsidered such as iterating a few times between the twoproblems within each time step. The accuracy improve-ment using such schemes was found to be insigni®cant.

2.4Failure criterionFailure may occur in one of two ways, over¯ow and rup-ture. Over¯ow occurs when water ¯ows over the dam.Rupture occurs when the breaking stress for the dammaterial is reached. The critical stress used here is19.65 MPa. This stress is obtained by using a simple rule ofmixtures to a rubber and nylon laminate (86% rubber,14% nylon) typically used in the construction of in¯atabledams.

2.4.1Determination of the failure envelopeFailure is governed by three parameters: the internal dampressure the ¯ood depth and the ¯ow Froude number. Two¯ood depths were chosen for this study, 2.0 and 3.0 meters.A serviceability envelope was produced for each depth.This envelope is the region of the internal pressure versus¯ood Froude number where the in¯atable dam does notfail due to either over¯ow or rupture. A similar envelopecould be developed for internal pressure versus ¯ooddepth, but was not done in this study.

The upper limit of the failure envelope correspondingto the internal pressure when the dam ruptures under

Table 1. In¯atable dam properties

Property Value

Dam design Initial radius 4.5 mInitial thickness 1.27 cmAnchoring type Double pinned

Material Density 100 000 kg/sq mPoisson's ratio 0.45Modulus of elasticity 103.4 MPaRupture stress 19.65 MPa

57

hydrostatic pressures was found by iteratively runningthe static ®nite element model. Different values for theinternal pressure were used until two values were ob-tained, one on each side of the rupture line, which onlyhad a two percent difference. The lower limit of thefailure envelope where the dam just over¯ows was foundby running the full ¯uid structure interaction model insimilar fashion as before while the Froude number washeld at a ®xed value. When the internal pressure re-quired to suppress the ¯ood exceeded the rupture value,the maximum ¯ood Froude number sustainable hadbeen reached, and the failure envelope was completelyde®ned.

3Numerical results

3.1Fluid responseFigure 5 shows a sample displacement pro®le for a typicalrun in which the dam does not fail in over¯ow or rupture.

As an alternative representation, Fig. 6 depicts the motionin a pro®le-temporal plot. As seen in these ®gures a bore ofwater is rejected to the left. Initially the ¯ood pushes thedam to the right and subsequently the dam oscillates backand forth. This oscillation causes small waves to occur ontop of the bore right next to the dam. As time progresses,the water height in front of the dam reaches a steady value.This height depends on the depth-based Froude number ofthe ¯ood. The primary contribution to the forces exertedon the dam is from the hydrostatic pressure increase due tothis increase in water level immediately in front of the dam.

Figure 7 is a sample free surface pro®le for a rigid damillustrating the formation of a jet of water where the freesurface contacts the dam. There is a jet of water pro-duced when the ¯ood ®rst strikes the dam in the rigidcase but not the ¯exible case. A jet occurs in manynonlinear free surface problems; see for example thewater entry problem analyzed by Hughes (1973). The jetdoes not form in the case of a ¯exible dam since thestructure deforms away from the jet and in a sensedamps it out.

Fluid - Structure interaction problem

Flood speed

Flexible inflatable damoscillates in response tothe impinging flood.

Upstreamboundarywith openboundaryconditions

Right hand boundaryis translated to the leftat the chosen floodvelocity (V) anddeforms with dam.

V

The fluid boundary isdescretized into straightline segments or panelsjoined together at nodes.Note: only some nodesare shown for illustration.

Free surface

Fluids towing tank simulator model

The flexible dam is comprisedof 100 ABAQUS S4R shell elements.Note: There are no elementson the bottom since it is not flexible.

Uniform pressure is applied acrosseach element. This pressure is the totalof the internal, hydrostatic, andhydrodynamic pressures.

Pinnedboundaryconditionsat edges

ABAQUS finite element dynamic analysis model

Fluid - structure coupling

The models are coupledat the contacting edge

which deforms withthe dam.

Boundary shape, andvelocity distribution foundusing ABAQUS model areapplied to the fluids model

Pressure distribution andfree surface contact point

found using the fluids modelare applied to ABAQUS model

Fluid - structure interaction problem issplit into two parts: the fluid flow, andthe forced dynamic structural response.These parts are solved by coupling a potentialflow towing tank simulator and an ABAQUSfinite element model

Flui

d flo

w p

robl

em

Forced dynamic

response of the structure

Dep

th

Fig. 4. Solving the ¯uid structure-interaction problem

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3.2Structural response

3.2.1Motion of the damFigures 5 and 6 show the motion of a dam for no over¯ow.As seen in the ®gure, the dam is pushed away from the¯ood and oscillates. The magnitude of the oscillationsdepends primarily on the internal pressure. As the internal

pressure is increased, the oscillations decrease in magni-tude, until they are hardly seen at all. At this point, theresponse of the ¯ood closely resembles the results for thecase of a rigid dam. A low pressure reduces the dam'sability to suppress the ¯ood.

When the internal pressure is brought below a certainvalue, the dam can no longer hold back the ¯ood, andover¯ow occurs. Figure 8 is a sample plot of an in¯atabledam starting to over¯ow. Over¯ow is seen in this study as

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The flood in this plot is running from left to right. A sufficient distance from the dam it has a depth of 3.0meters and Froude number (non-dimensional velocity) of 0.2. The dam is 4.5 meters tall and has anintemal pressure of 1.20 meters of water. After startup, the flood pushes the dam to the right.The floodwaters pile up in front of the dam and a left running bore of water is produced.

Having rebounded too far forward, the dam recedes once more. After a suffidently long time, the freesurface in front of the dam reaches a steady height. Some low amplitude waves are generatedduring the oscillation of the dam in a similar fashion as a flap type wavemaker.

The dam, having receded too far “sling shots” back to the left. A small wave is produced on the freesurface in front of the dam during this motion.

Free survace profile (Fn = 0.20, flood depth = 3.0 m, pint = 1.20 m H O)2

Fig. 5. Sample displacementpro®le for an in¯atable dam ina ¯ood

59

the point where the thin region of water over the damcollapses. When this occurs, the nodes on the free surfaceand the dam touch and break down the numerical com-putations. The over¯ow pressure found here will be con-servative as the dam may actually be barely able to containthe ¯ood after the numerical computations break down.

Figure 9 is a plot of the trace of several nodes on thesurface of the dam during a typical run. The node on theupstream side of the dam, at about the initial water level,displaces down and away from the ¯ood, then returns on

nearly the same path. The node on the top of the damfollows an orbital path such that it does not retrace its pathas it ``bounces back''. The node on the downstream side ofthe dam displaces forward and backward on about thesame path.

Figure 10 is a plot of the motion of the top node as afunction of time. As seen in the plot, the motion in the xdirection appears to be almost sinusoidal. For a ¯ooddepth of two meters, ¯ood Froude number of 0.3, and aninternal pressure of two meters of water, the period of this

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epth

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Fig. 6. Sample displacement ± timepro®le plot of the free surface and anin¯atable dam

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The flood in this plot is running from left to right. A sufficient distance from the dam it has a non-dimensionaldepth of 1.0 and a Froude number (non-dimensional velocity) of 0.20. This corresponds to a flood depthof 3.0 meters impinging on a 4.5 meter tall dam with a uniform upstream velocity of 1.09 m/s. The dam is arigid semicircle. During the begining of the flood the water piles up in front of the dam and a slug or bore ofwater is rejected to the left.

As time progresses the flood reaches a steady height over some distance in front of the dam. This distanceincreases over time, as the bore of water continues to the left . Note that there are fewer waves for a rigiddam than for the flexible one.

Fig. 7. Plot of the free surfaceresponse for a rigid dam

60

motion is roughly constant at about 0.377 seconds. Themotion in the y direction is not as uniform. This showsthat there is more than one mode being excited. The am-plitude of the motion of a point on the dam is damped outduring the run due to wave generation on the free surface.

3.2.2Stresses in the damFigure 11 is a plot of the static stress (upstream ¯oodspeed = 0) in the dam for a low in¯ation pressure. The

tensile stress is roughly constant over the surface of thedam. For an internal pressure of 1.75 meters of water,and a water depth of 3.0 meters the mean circumfer-ential stress is 75 KPa. The mean transverse stress is35 KPa. The maximum stress due to bending occursabout one meter from the bottom on the ¯ood side ofthe dam. For this case, the maximum stress due to acombined state of bending and tension is about340 KPa. For low pressures the bending stress is signi-®cant.

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The flood in this picture is running from left to right. A sufficient distance from the dam it has a non-dimensionaldepth of 1.0 and a Froude number (non-dimensional velocity) of 0.30. This corresponds to a flood depth of threemeters impinging on a 4.5 meter taIl dam with a uniform upstream velocity of 1.63 m/s and an internal pressure of1.75 meters of water During the beginning of the motion the water piles up in front of the dam as it recedes to theright, in the flow direction.

As time progresses, the dam is pushed further to the right. As it is pudhed over the height decreases,and eventually the flood water starts to go over the top of the dam.

Fig. 8. Sample displacement pro-®le for an in¯atable dam duringover¯ow

Hydrostatic shape(Fn = 0, time = 0)

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Trace of several nodes on the dam surface with time(Fn = 0.30, depth = 3.0 m, pint - 2.0 m H O)2

Fig. 9. Trace of several nodeson the surface of an oscillat-ing in¯atable dam

61

Figure 12 is a plot of the static stresses in the dam forthe rupture pressure in a 3.0 meter deep ¯ood. For highpressures, closer to the rupture pressure, the bendingstress is insigni®cant. For an internal pressure of 4.85meters of water (the rupture pressure for this depth) and awater depth of 3.0 meters, the mean circumferential stressroughly equals the maximum stress (through the thick-ness). The mean circumferential stress is 19.5 MPa, andthe mean transverse stress is 8.82 MPa.

Figure 13 contains sample results for the dynamic stress¯uctuation with time at a node on the top of the dam. Thestresses at other locations in the dam are qualitativelysimilar to this sample. It is an interesting to note that the¯ood actually lowers the stresses in the dam material. AsFig. 13 indicates, as time increases, the mean stress drops.As a result, the highest stress occurs near the beginning of

the run, and the dynamic stress ¯uctuation is only about10% of the total stress. Therefore, if the dam is going torupture, it will most likely occur at the beginning understatic loading. This may be explained by the fact that thewater depth in front of the dam increases due to the ¯ood.The increasing hydrostatic pressure outside the dam par-tially counteracts the pressure inside the dam. This resultsin decreased stresses in the dam.

3.2.3The operational envelopeFigures 14 and 15 are plots of the serviceability regions for¯ood depths of two and three meters, respectively. Acomparison of these two plots reveals that for a lower ¯ooddepth, a faster moving ¯ood can be suppressed. In addi-tion, the rupture pressure for a lower ¯ood depth is

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Fig. 10. The motion of theuppermost node of an oscil-lating in¯atable dam as afunction of time

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Maximum circumferential stress isdue to a combined state of bendingand tension.

Static stresses along the dam surfacedepth = 2.0 m, pint = 1.75 m H O2

Fig. 11. Plot of the static stress in the dam for a low in¯ationpressure

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Nearly coincident curves for thetensile stress, and the maximumcircumferential stress

Static stresses along the dam surfacedepth = 3.0 m, pint = 4.85 m H O2

Fig. 12. Plot of the static stresses in the dam for an in¯ationpressure that would cause the dam to rupture

62

actually lower than that for a higher ¯ood. This is due tothe partial cancellation of external and internal pressures.A higher depth counteracts more of the applied internalpressure, and actually stabilizes the dam against rupture.From Figs. 14 and 15, it may be concluded that existingin¯atable dams are quite effective in suppressing ¯oods fora relatively wide range of ¯ood velocities.

4ConclusionsThe effectiveness of a ¯exible in¯atable dam to suppress¯oods in emergencies was studied numerically. Suchstructures have been used for water storage and irrigationbut not to control ¯oods. To this end, the dynamic, non-linear response of an in¯atable dam to a ¯ood was studiedfor a range of ¯ood depths and ¯ood speeds. The nu-merical analysis was carried out in the time domain. A

fully nonlinear Eulerian-Lagrangian formulation was usedto simulate the ¯uid ¯ow. The dam is modeled as an elasticshell in¯ated with air and simply supported from twopoints. The ®nite element method was employed to de-termine the dynamic response of the structure usingABAQUS with a shell element.

The dimensions and material properties of the structureused in the present study are representative of existingstructures. No effort was made in the present work to varythe material properties and the dimensions of the structureand optimize the structural performance for a ®xed cost.The serviceability regions for ¯ood depths of two and threemeters, respectively were obtained as functions of the in-ternal pressure. If this pressure is too low, the water willover¯ow the dam. If it is too high, the dam will rupture.From Figs. 14 and 15, it may be concluded that existingin¯atable dams are quite effective in suppressing ¯oods fora relatively wide range of ¯ood velocities.

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Dynamic stress response for an internal pressure of 2.0 m of water, flood depth of 3.0 m,and flood Fn of 0.30

Fig. 13. Dynamic stresses at apoint on the top of the dam

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Serviceability region(Dam does not fail)

Fig. 14. Serviceability envelope for a ¯ood depth of two meters

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Fig. 15. Serviceability envelope for a ¯ood depth of three meters

63

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