numerical model…..graviry dams

8/2/2019 Numerical model..graviry dams

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Numerical Model Validation 233

NUMERICAL MODEL VALIDATION FOR LARGE CONCRETE

GRAVITY DAMS

Florian Scheulen1

Nicolas Von Gersdorff2

Ziyad Duron,

3

Ph.D.Mike Knarr,4

P.E., S.E.

ABSTRACT

The interaction effects between a large concrete gravity dam, its foundation, and

impounded reservoir have been investigated using 2-D and 3-D finite element

representations validated against field measurements. Field tests conducted on the damprovided acceleration profiles across the dam crest and along the dam-foundation

interface, and were used to assess material properties in the dam and in the foundation.

Field measurements of hydrodynamic pressures acquired along the upstream dam face

were used to evaluate the relative accuracy of various techniques for representing thereservoir water in the numerical model of the dam-foundation-reservoir system.

Techniques for representing the reservoir water included the Westergaard added massapproach, the RSVR2 approach, fluid elements modeled as plane strain elements with the

properties of water and acoustic elements. Comparisons of measured and computed

acceleration and hydrodynamic pressure frequency responses were obtained that illustrate

the advantages and limitations associated with each modeling approach. The paperprovides relevant details associated with the field tests on the dam, presents an overview

of the numerical models, and discusses the assumptions used in developing the reservoir

representations. Frequency responses are compared to highlight model performance.These results will be used to validate numerical models of the dam-foundation-reservoir

system in support of an ongoing risk-based performance evaluation for this dam.

INTRODUCTION

An accurate finite element model is crucial for most risk-based dam analyses. This paperdemonstrates how field data can be used to validate the accuracy of a model. Various

methods of modeling the reservoir were applied to 2-D and 3-D models of a large

concrete gravity dam, Big Creek Dam No. 7, and compared to hydrodynamic pressurevalues acquired using hydrophones. Acceleration profiles along the crest and at the dam-

foundation interface were used to enhance the models ability to accurately reproduce

observed behavior, and the models ability to predict observed hydrodynamic pressureresponse was ultimately used as the basis for validating model response behavior. An

1 De Pietro Research Fellow, Department of Engineering, Harvey Mudd College, Claremont, CA 91711,

[email protected] Structural Engineer, Southern California Edison Company, San Dimas, CA 91773, [email protected] Professor of Engineering, Department of Engineering, Harvey Mudd College, Claremont, CA 91711,

[email protected] Principal Structural Engineer, Southern California Edison Company, San Dimas, CA 91773,

[email protected]


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Collaborative Management of Integrated Watersheds234

overview of the field investigation is presented.

Big Creek Dam No. 7 was completed in 1951. It is located on the San Joaquin River in

Fresno County, California and impounds Redinger Lake which has a surface area of 465

acres. Figure 1 shows an upstream view of the dam. The dam is 250 feet high, with a

compound downstream slope of 0.75 horizontal to 1.0 vertical on the upper portion,flattening to 0.83 horizontal to 1.0 vertical near the bottom. The dam is 875 feet long and

consists of 19 monoliths.

Figure 1. Big Creek Dam No.7 profile

EXPERIMENTAL TESTING

Forced vibration tests were conducted on Big Creek Dam No. 7 in the summer of 2008. A

large eccentric mass vibrator, or shaker, was employed to induce steady-state responses

in the dam over a 20 Hz range at force levels approaching 20,000 lbf. The frequency

spacing was selected to be 0.05 Hz. The mass vibrator and the associated force curves are

shown in Figure 2. The blue line indicates the force curve for the large weights and thered line for the small weights.

4 8 12 16 200

4,000

8,000

12,000

16,000

20,000

Frequency (Hz)

Force(lbf)

Shaker Force Curves

Figure 2. The shaker system and force curves used for vibration testing.

Hydrodynamic pressure responses in the reservoir along the upstream face of Monolith 6were acquired using a hydrophone instrument array. The array consists of 8 solid-state


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hydrophones spaced 50 ft apart along an 800 ft long Kevlar reinforced cable. The

hydrophones are Halliburton Geophysical Services Model 12 and contain a piezoelectriccrystal which produces a voltage proportional to hydrodynamic pressure. Hydrophone

sensitivity averaged 0.7 V/psi. The reservoir depth at the measurement location was less

than 150 ft, so the hydrophones were looped such that the spacing was reduced to 25 ft,

allowing for a total of 6 pressure measurements to be made.

Accelerations on the crest of the dam and along the foundation interface were acquiredusing Q-Flex Model QA-700 and QA-750 accelerometers manufactured by Honeywell.

These accelerometers incorporate a servo-force-balance design to produce a current

output which is proportional to surface acceleration, which allows extended cable lengths

to be used throughout the dam without loss of signal quality. Accelerometer sensitivitiesranged from 10.7 V/g to 12.3V/g for the tests at Dam No.7.

Analog signals were band-pass filtered using a 2-pole high-pass Butterworth with cutoffat 1 Hz and a 4-pole low-pass Butterworth with cutoff at 30 Hz. Amplification gains

were varied depending upon signal strength between 10 and 1000 to fill the dynamicrange of the A/D converter. These signals were digitized at a rate of 1000 samples persecond to achieve satisfactory signal quality while guarding against unwanted aliasing

effects. A sample steady-state time history of hydrodynamic pressure data is shown in

Figure 3. This is representative of the signal quality observed during testing, for bothhydrodynamic and acceleration measurements.

0 0.2 0.4 0.6 0.8 1-0.02

-0.01

0

0.01

0.02

Time (sec)

Pressure(psi)

Sample Steady-State Time HistoryDepth 67.5 ft

Figure 3. Sample steady-state time history of hydrodynamic pressure at 67.5 ft below thewater surface.


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FINITE ELEMENT MODELING

2-D Single Monolith Finite Element Model

A 2-D finite element model of the 7

th

monolith, the tallest non-spillway monolith, whichis representative of non-spillway blocks 5 to 7 and 11 to 13, was developed. It was

modeled using SAP2000, which is a commercially available general purpose finite

element package (Ref 1).

The model includes the dam and portions of the impounded reservoir and underlying

foundation. A 2-D mesh is shown in Figure 4.

Figure 4. Mesh of 2-D model of the 7th

monolith.

The model of the gravity dam consists of four-node plane strain elements, which are

restrained out-of-plane, thus supporting two translational degrees of freedom and one

rotational degree of freedom.

The foundation model consists of the same elements and extends approximately one dam

height upstream, downstream, and below the dam. The bottom and side boundaries of the

foundation were modeled using a fixed condition at the base and translations fixed

laterally but free to move vertically at the sides. The foundation model was assumed asmassless.

The reservoir was modeled using various methods as will be detailed in later sections.


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3D Full Dam Finite Element Model

A 3-D model of the dam-foundation-reservoir system was constructed and analyzed using

Abaqus (Ref 2), a commercially available general purpose finite element package. The

model was constructed for the purpose of evaluating its ability to reproduce observed

behavior. It incorporates some major features of the dam system while others, such as thespillway gates, are accounted for by lumped masses or parameter variation. Features not

modeled include the spillway radial gates, the deck over the spillway section, thepenstocks or outlets, and the gallery in the dam. The model of the dam consists of eight-

node brick elements whose material properties were based on previous core test results.

The model of the foundation was contoured and shaped using the outline of the dam to

extend away from the dam in the stream direction a distance equal to the dam height.The model of the reservoir was placed at an elevation 15 ft below the crest, the

approximate level of the reservoir during the actual forced vibration test period.

The dam model incorporates 17 monoliths which are separated by vertical joints. The

joints are not keyed, but lower portions of the tallest monoliths are grouted. To accountfor the condition of the joints in the dam, and to provide capability within the model toevaluate the influence of joint closure or opening on dam response, the joints in the dam

were modeled as a thin (1 in wide) column of elements between monoliths. The column

allowed material property parameters to be adjusted so that fully closed (high modulus,nominal concrete mass density) or open (low modulus, low mass density) effects of joint

behavior on dam response could be evaluated. Modeling actual joint behavior in a large

dam, particularly under strong motion loading conditions typically requires sophisticated,

non-linear modeling techniques. However, for the studies and comparisons of the low-level response behavior of Dam No. 7, a linear elastic model capable of adjusting relative

joint stiffness and mass to account for open or closed joint behavior was deemedsatisfactory.

The reservoir geometry included in the model is shown in Figure 5.

Figure 5. Reservoir model geometry (images not drawn to scale).

The reservoir model includes a sediment layer along the floor, known to be present in thereservoir at Dam No. 7, and indicated in brown (left image, Figure 5). The sediment

layer begins 40 ft upstream from the dam and slopes upward to the measured elevation

within 15 feet of the bottom of the original reservoir. This sedimentation profile was

Water only:


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determined to give the best results. Sediment thickness varies from 4 ft to 27 ft within

the reservoir.

A complete mesh of the model is shown in Figure 6.

Figure 6. Meshed 3-D finite element model of Dam No. 7.

The dam and foundation elements are eight-node linear solid bricks, and the reservoirelements are eight-node linear acoustic elements. Interactions along the dam-reservoir

and foundation-reservoir interfaces are selected to ensure that only normal motions of the

dam and foundation result in pressure changes in the reservoir. The acoustic element waschosen to facilitate comparisons against measured steady-state hydrodynamic pressures,

and incorporates the assumption of compressible, inviscid fluid flow.

The foundation boundaries are fixed everywhere except along the foundation surface atthe dam crest elevation. Reservoir water boundary conditions consist of setting the

surface acoustic element nodes to zero pressure, consistent with actual atmospheric

conditions in the reservoir. The boundary at the far upstream face of the reservoir modelwas evaluated under both prescribed and unprescribed conditions. Using the acoustic

element formulation, a nonreflecting boundary can be specified in terms of an acoustic

impedance to define a termination. The impedance can be user defined or associatedwith a range of geometries available in Abaqus. For the comparisons shown here, an

acoustic termination associated with a planar geometry was selected.


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PARAMETER STUDY

A parameter study was conducted on the 3-D model. Material property values included in

the dam-foundation-reservoir model are listed in Table 1. Included are values for the

concrete in the dam, the rock in the foundation, and the water and sediment in the

reservoir. The joint properties listed in Table 1 were arrived at after a series ofeigenanalyses in which model behavior, characterized in terms of resonant frequencies

and response shapes, was compared against measured behavior from the forced vibration

tests.

Table 1. Material properties used to evaluate dynamic response behavior of Dam No. 7.

Property Measured Model

Concrete (Dam)

Density 150 pcf 150 pcf

Youngs Modulus 3.3 Mpsi 4.2 Mpsi

Poissons Ratio 0.2 0.2

Foundation (Rock)Density 150 pcf 0.150 pcf

Youngs Modulus 3 Mpsi 2.5 Mpsi

Poissons Ratio 0.25 0.25

Monolith Joints

Density N/A 0.15 pcf

Youngs Modulus N/A 7.3 Kpsi

Poissons Ratio N/A 0.2

Water (Reservoir)

Density 62.4 pcf 62.4 pcf

Bulk Modulus 315 Kpsi 181 Kpsi

Water (Sediment)

Density 125 pcf 125 pcf

Bulk Modulus N/A 218 Kpsi

To arrive at these values, an iterative approach was used in which the material properties

were varied and the numerical frequency response computed and compared to theexperimental data. The two aspects that were targeted for this comparison were the

location and magnitude of the resonant peaks along the dam.

Computed frequency response functions were obtained from a steady-state harmonic

analysis using the model in which both a direct integration and modal solution approach

were evaluated. The direct integration approach is assumed to provide added accuracy in

terms of evaluating the full response model behavior, as compared to results from amodal approach which truncates the models frequency characteristics included in the

response calculations. To assess the relative accuracy of the direct integration and modal

solutions for Dam No. 7, analyses using both approaches were completed and resultsfrom each are included in the comparisons of acceleration responses. Damping

associated with each approach differed slightly Rayleigh damping was used to ensure

5% of critical at the 1st

and 3rd

modes, and damping in the amount of 5% of critical was


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used for all modes included in the modal solution. Fifty modes were used in the modal

solution approach representing behavior between 3-20 Hz, chosen to match the frequencyrange during test. The magnitude and phase of the frequency response function for

Monolith 7 is shown in Figure 7.

0 5 10 15 200

0.2

0.4

0.6

0.8

1

Frequency (Hz)

Resp

onseMagnitude(g/Mlbf)

Numerical and Experimental FRF ComparisonMonolith 7

Direct

Modal

Experimental

0 5 10 15 200

50

100

150

200

Frequency (Hz)

ResponsePhase(

degrees)

Numerical and Experimental FRF ComparisonMonolith 7

Direct

Modal

Experimental

Figure 7. Magnitude (top) and phase (bottom) of the Acceleration Frequency Response at

Monolith 7.

The comparison shown is considered to be quite good. The models frequency behaviorbelow 6 Hz compares well against observed behavior even though the character of the

response is pseudo-dynamic at best, and the agreement suggests that the stiffness-mass

ratio of the model in this frequency range is reasonable. The models behavior in the 9-14 Hz frequency range is characterized by single and closely coupled peak response. The

modal solution provides a better match with the experimental result based on the apparent


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modal coupling between 10-13 Hz. The direct solution response does not indicate the

same degree of coupled behavior. Nonetheless, the overall magnitudes in these responsesin this frequency range are consistent. Between 14-20 Hz, the comparisons remain fairly

consistent in terms of magnitudes and trends, but details associated with individual

resonant peaks vary throughout this frequency range.

The agreement in the phase response curves is also considered to be quite good,

especially when the complex nature of the interaction effects between the dam-foundation, dam-reservoir, and foundation-reservoir are considered.

Foundation flexibility was measured during the forced vibration tests at Dam No. 7 by

measuring vertical acceleration response at the heel and toe of the 7th

and 11th

monoliths.This test is similar to those previously conducted on large concrete gravity dams

including Folsom Dam and San Vicente Dam, where flexibility was characterized by the

ratio of heel-to-toe response (magnitude) that exhibited a 180 degree (phase) difference.For those dams, foundation flexibility was reported using a single ratio at the particular

frequency where 180 degree response was indicated.

The numerical model studies of Dam No. 7 have allowed a broader study of foundation

flexibility. Shown in Figure 8 is a comparison of measured and predicted (model)

frequency response function defined as the ratio of vertical heel-to-toe response atMonolith 7.

6.5 7 7.5 8 8.5 90

1

2

3

4

Frequency (Hz)

RelativeAccelerationR

esponse

Numerical and Experimental FRF ComparisonHeel-to-Toe Monolith 7

Direct

ModalExperimental


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6.5 7 7.5 8 8.5 9 9.5-350

-300

-250

-200

-150

-100

Frequency (Hz)

RelativePhase(de

grees)

Numerical and Experimental FRF ComparisonHeel-to-Toe Monolith 7

Direct

Modal

Experimental

Figure 8. Relative acceleration magnitude (top) and phase (bottom) of the heel-to-toe at

Monolith 7.

The best match between model and observed relative behavior, where magnitudes are ofthe same order and phase is near 180 degrees (denoted by a dashed line in Figure 8) ,

occurs in the vicinity of 8 Hz. Outside this narrow range, however, the comparison of

relative magnitude is poor, even though the phase comparison appears to be slightly

better (below 8 Hz).

A comparison of magnitude and phase heel-to-toe rocking behavior defined at 7.75 Hz,

where phase is at or near 180 degrees, is provided in Table 2. These values are takendirectly from the predicted and measured vertical responses in 7

thand 11

thmonoliths

where near out-of-phase motion (rocking) at each base is observed.

Table 2. Foundation flexibility indicative of near out-of phase rocking base response.

Monolith 7 Monolith 11

Method Phase(degrees) Magnitude Phase (degrees) Magnitude

Model/Direct -205 1.357 -134 1.158

Model/Modal -193 0.842 -146 1.051

Experimental -192 0.467 -173 3.226

These foundation flexibility studies have been used to adjust the elastic modulus used in

the foundation finite element models.

RESERVOIR MODELING METHODS

Westergaard

Westergaard added mass was incorporated into the finite element analysis by applying amass to each upstream node, scaled by the tributary area around that node. The added

mass is equal to:


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(1)where b is added mass, H is the depth of the reservoir at the node, y is the depth of the

node below the water, is the density of the water, g is the acceleration due to gravity,and A is the tributary area.

John Halls RSVR2

RSVR2 is John Halls upgraded version of RSVR. It calculates added mass similar to the

Westergaard method as well as additional hydrodynamic forces on the dam which arisefrom accelerations of the floor and sides of reservoir. The output of the program was

applied to the 2-D model as added mass and force multiplied by earthquake acceleration.

Fluid Elements

The water was modeled as a fluid using four-node plane strain elements in the 2-D

model. For a 3-D model eight-node brick elements can be used in a similar way. Thewater mass is included to capture hydrodynamic effects, but the weight of the water was

not included because the hydrostatic pressure was modeled using pore pressure.

Essentially, the gravitational pull of the water was set to zero. The properties assigned tothe fluid elements were:

Table 3. Fluid element properties

Density 62.4 pcf

Shear modulus 63.24 psi

Modulus of elasticity 189.7 psi

Wave speed 4720 ft/sec

Poissons ratio 0.4999

Bulk modulus 3.16 Mpsi

Using plane strain elements to represent the water adds a large number of extraneous

modes to the analysis. To minimize this effect, Ritz vectors were used in lieu of thestandard eigenvectors. The first mode of the dam was found to be at the calculated 43

rd

mode of the analysis. All preceding modes were found to be water modes with no

significance to the analysis.

Boundary Conditions: The upstream boundary of the water was restrained laterally but

free to move vertically. Gap elements were introduced between the foundation and water

and between the dam and the water to keep the water in contact with the dam andfoundation. The gap elements allowed only compression to be transmitted from the water

to the dam and foundation.


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Acoustic Elements

The following governing relationships apply to acoustic elements, whose single degree of

freedom is pressure:

(2)where is the dynamic pressure, is the spatial position of the fluid particle, is the

fluid particle velocity, is the fluid particle acceleration, is the volumetric drag, and

is the density.

The constitutive relationship for the acoustic fluid is given by:

(3)where is the dynamic pressure, is the bulk modulus, and is the volumetric strain.Additional details concerning the formulation of the acoustic finite element in Abaqus arefound in the Abaqus/Standard Users Manual (Ref 2).

COMPARISON OF RESULTS

Resonance Frequency Comparison

The eccentric mass shaker forces were applied at the crest of the 2-D and 3-D models.

Figure 9 - Figure 12 below show a comparison of Westergaard, fluid element, and

acoustic element generated resonances to the hydrophone data. Comparisons are made atsix different reservoir depths along the upstream face of Monolith 6, where thehydrophones were located. Distances are from the top of the reservoir surface.

Using the equation:

(4)where C is the wave speed of water and H is the height of the reservoir, produces a rough

estimate of the resonance frequency of the reservoir. At the 7th

monolith of Big CreekDam No. 7 the height of the reservoir is 182 feet, making the reservoir resonance

estimate 6.5 Hz. The hydrophone data shows reservoir resonances around 5 and 6.8 Hz.Westergaard is close to the first resonance at 4.6 Hz, but completely misses the secondresonance. The results of the RSVR2 output are almost identical to Westergaard and have

been omitted. The fluid element resonances are at 4.7 and 6.8 Hz. The first and second

resonances of the acoustic elements are at 4.8 and 6.9 Hz. The magnitude of the first

resonance is around 0.2 (psi/million lbf), which is close to the hydrophone data. The fluidelement and Westergaard magnitudes are significantly higher because the shaker forces

are being applied to a single monolith instead of being distributed across the entire dam.


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3 4 5 6 70

0.1

0.2

0.3

0.4

Frequency (Hz)

Hydrophone Data

Pressure/Force(psi/millionlbf)

17.5 ft

42.5 ft

67.5 ft

92.5 ft

117.5 ft

142.5 ft

Figure 9. Hydrophone pressure results.

3 4 5 6 70

5

10

15

Frequency (Hz)

Westergaard (added mass)

Pressure

/Force(psi/millionlbf)

17.5 ft

42.5 ft

67.5 ft

92.5 ft

117.5 ft

142.5 ft

Figure 10. Westergaard pressure results.


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3 4 5 6 70

2

4

6

8

10

12

Frequency (Hz)

Fluid Elements

Pressure/Force(psi/millionlbf)

17.5 ft

42.5 ft

67.5 ft

92.5 ft

117.5 ft

142.5 ft

Figure 11. Fluid element pressure results.

3 4 5 6 70

0.2

0.4

0.6

0.8

1

Frequency (Hz)

Acoustic Elements

Pressure

/Force(psi/millionlbf)

17.5 ft

42.5 ft

67.5 ft

92.5 ft

117.5 ft

142.5 ft

Figure 12. Acoustic element pressure results from modal solution approach.

Stress Comparison

The stress results of the 2-D model were used to compare Westergaard, RSVR2, fluid

elements, and output from a program developed by Chopra (Ref 3). The time history usedin finite element analysis was from the Imperial Valley record. All stress results weretaken at the heel of the monolith, where stresses are generally highest in an earthquake.

Heel cracking due to high stresses can increase uplift pressures significantly,

undermining the structures stability.


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As seen in Table 4, Westergaard was found to be very conservative with results 15%

greater than Chopra, while the fluid element results were within 6% of the Chopra results.RSVR2 results were nearly identical to Westergaard.

Table 4. Stress results from the 2-D model.

Chopra Westergaard Fluid Elements(psi) Time History (psi) Time History (psi) Response Spectrum (psi)

187 215 177 137

CONCLUSIONS AND FURTHER STUDY

Experimental results can be used to calibrate the parameters of a numerical model to

achieve desired performance. Model performance based on the comparisons presented in

this paper suggests that the models constructed of Dam No. 7 achieve a good measure ofcorrelation with observed behavior below 16 Hz. This is particularly the case in the

modes that correspond to the crest and upper portions in the dam, where reasonably goodagreement is achieved.

Various reservoir modeling methods have been compared:

The Westergaard approach is quick and easy to implement. With fewer elements it also

requires the least amount of computing power. It generally produces overly conservative

stress results and does not match all the resonances seen in the hydrophone data, but it isstill sufficient for a preliminary analysis.

The RSVR2 method includes additional forces that may make it more accurate than the

Westergaard method but still has the same shortcomings. It also takes considerably moretime to implement.

The model incorporating fluid elements matches the hydrophone data well and producesgood stress results. A large number of water sloshing modes were generated that made

it more difficult to analyze the reservoir-dam system since they would crowd out modes

with large participation ratios.

The acoustic element method produces results that are an excellent match with the

hydrophone data. Matching the resonances seen in the hydrophone data is of great benefit

in developing confidence in the models accuracy to represent the real behavior of the

dam in response to ground motion.Water sloshing modes are not generated becauseacoustic elements do not translate.

More sophisticated reservoir modeling techniques such as coupled Eulerian-Lagrangian

systems could yield even better results, but are more difficult to implement and

necessitate considerably more computing power.


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REFERENCES

1. SAP2000, Version 12.0.2, Computers and Structures, Inc., 2009.2. Dassault Systemes Simulia Corp. Section 21.3.1 Abaqus/Standard Users

Manual. Version 6.8, 2008.

3. Simplified Earthquake Analysis of Concrete Gravity Dams, Fenves and Chopra,Journal of Structural Engineering, Vol. 113, No. 8, August 1987, pp. 1688-1708.

numerical model…..graviry dams

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