Seismic Sloshing in a Horizontal Liquid-Storage Tank

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To appear in the Journal of Structural Engineering International

Seismic Sloshing in a Horizontal Liquid-Storage Tank

Praveen K. Malhotra, Parag Nimse and Michael Meekins

Summary

A horizontal water storage tank was analyzed for seismic shaking at the 5th level of the ITER

Tokamak Complex in France. The objectives were to: (a) estimate the seismic forces in the tank;

(b) calculate the sloshing response of the tank; (c) determine if baffles are needed to control

sloshing; and (d) evaluate the possibility of using a single fixed support in the longitudinal

direction to allow free thermal expansion of the tank. Approximate conservative analysis

predicted very high sloshing wave and seismic forces in the tank. Fluid-structure interaction of

the tank showed that only about 28% of the liquid moves with the tank wall and generates

seismic forces in the longitudinal direction. The remaining 72% of the liquid sloshes near the

free-surface and does not generate significant seismic forces. The sloshing wave is not high

enough in the longitudinal direction because the fundamental sloshing mode is not excited due to

its very low natural frequency. Hence baffles are not needed to control sloshing. The seismic

force was low enough that a single fixed support can resist the entire seismic force in the

longitudinal direction.

Keywords: sloshing, seismic, tank, liquid-storage, fluid-structure interaction, finite element

analysis



Introduction

The seismic response of a liquid-storage tank can be understood by visualizing that the liquid in

the tank is divided into two parts: (a) convective and (b) impulsive [1-4]. The convective liquid

sloshes independently of the tank wall and the impulsive liquid moves with (and deforms) the

tank wall. The relative proportions of the convective and impulsive liquid depend on the ability

of the tank to confine the liquid. Greater the confinement, greater is the impulsive liquid and

smaller the convective liquid. The natural frequency of the convective vibration is low, but the

natural frequency of the impulsive vibration is high. As a result, the convective liquid

experiences low accelerations and high deformations, but the impulsive liquid experiences high

accelerations and low deformations during seismic shaking. The convective liquid causes the

free-surface to move up-and-down (slosh) but it does not apply significant pressures on the tank

wall. The impulsive liquid applies pressures on the tank wall and induces seismic forces in the

supports. The up-and-down movement of the free-surface can be reduced by adding baffles in

the tank, but the baffles also confine the liquid and increase the impulsive mass hence the

seismic forces. Therefore, a trade-off exists between the free-surface sloshing and the seismic

forces in the tank.

The International Thermonuclear Experimental Reactor (ITER), currently under construction in

France, has a Vacuum Vessel Pressure Suppression System (VVPSS). The VVPSS includes a

horizontal tank filled with water to condense steam resulting from an in-vessel coolant leak in

the vacuum vessel. The condensed steam can cause the temperature of the water to rise from

room temperature to 80 C. The temperature rise will cause significant stresses in the tank shell

if the tank is not able to expand freely in the longitudinal direction.

The tank is required to remain operational following the safe-shutdown earthquake (SSE) with a

mean return period (MRP) of about 10,000 years. The inlet pipes from the top of the tank, also

known as quenchers, are required to remain immersed in water during the SSE event. A pre-

design study was carried out to:

1. Estimate the sloshing response of the tank and assess the need for baffles to control sloshing;

and

2. Estimate seismic force in the tank and evaluate the possibility of a single fixed-support to

allow free thermal expansion of the tank in the longitudinal direction.



Tank Size, Location and Geometry

The VVPSS tank is located at level L5 of the ITER Tokamak Complex. The Tokamak Complex

is base-isolated in both horizontal directions with elastomeric bearings [5]. Figure 1 shows a

longitudinal section of the tank. The inner diameter of the tank is 5.8 m. The tank is 46.26 m

long and it has a uniform wall thickness of 30 mm. The two ends of the tank have convex walls

of 5.22 m radius. The tank rests on 9 saddle supports (not shown in Figure 1). The water level in

the tank is 19 cm above the half point. The mass of the empty tank, including saddle supports, is

276,000 kg and the mass of the liquid in the tank is 675,000 kg. The total mass of the liquid-

filled tank is 951,000 kg. The inlet pipes from the top of the tank (quenchers) are immersed

2.69 m into water (Figure 1); they are along the middle section of the tank in the longitudinal

direction. The material of the tank is stainless steel 304/304L with following properties: modulus

of elasticity = 193 GPa, Poisson’s ratio = 0.29, and mass density = 8,030 kg/m3.

Figure 1. Longitudinal section of the VVPSS tank at ITER facility (all dimensions are in

millimeters).



Seismic Floor Motions

The SSE level floor motions were provided by ITER in terms of the floor response spectra.

According to ITER load specification guidelines [6], the damping for convective (sloshing)

modes of vibration is 0.5% of critical and the damping for impulsive (structural) modes of

vibration is 3% of critical. The 0.5% damping response spectra were directly provided by ITER.

The 3% damping response spectra were generated by logarithmic interpolation between the 2%

and 5% damping response spectra in accordance with ITER load specification guidelines [6].

Figures 2, 3 and 4 show the 0.5% and 3% damping response spectra in the longitudinal,

transverse and vertical directions of the tank. In these plots: the peak pseudo-velocity PPV is

shown along the vertical axis; the peak pseudo-acceleration PPA is shown along the 45 axis

(clockwise); the peak deformation PD is shown along the -45 axis (counter-clockwise); and the

natural frequency F is along the horizontal axis. The PPA is shown in units of g = acceleration

due to gravity = 9.81 m/s2. PPA, PPV and PD are relates to each other as follows:

PPA/ 2 F PPV PD 2 F (1)

Figure 2. SSE level floor response spectra for 0.5% and 3% of critical damping in the

longitudinal direction of the tank.



The floor response spectra in the longitudinal (Figure 2) and the transverse (Figure 3) directions

show two prominent humps (peaks). The first hump is around a frequency of 0.5 Hz. This

corresponds to the fundamental translational frequency of the base isolated Tokamak Complex.

This could affect the sloshing response of the VVPSS tank. The second hump in the response

spectra is around a frequency of 7 Hz. This corresponds to the rocking mode of vibration of the

Tokamak Building due to axial deformation of the base isolators [5]. This could affect the

impulsive (structural) response of the VVPSS tank. The Tokomak complex is not isolated in the

vertical direction. The response spectra of vertical motion (Figure 4) show a hump around a

frequency of 7 Hz which again corresponds to the axial deformation of the elastomeric base

isolators [5]. This could affect the impulsive response in the vertical direction of the VVPSS

tank.

Figure 3. SSE level floor response spectra for 0.5% and 3% of critical damping in the

transverse direction of the tank.

The response spectra plots also show the peak values of floor acceleration and displacement in

the three directions. The floor acceleration in the vertical direction is about five times the floor

acceleration in the horizontal directions; the floor displacement in the vertical direction is about



one-third the floor displacements in the horizontal directions. Thus the vertical motion is richer

in high frequencies and the horizontal motion is richer in low frequencies.

Figure 4. SSE level floor response spectra for 0.5% and 3% of critical damping in the vertical

direction of the tank.



Conservative Analysis

If the tank were small and/or the input motions were not dominated by a few frequencies, it

could have been conservatively assumed that the response acceleration of the tank-liquid system

equals the peak of the 0.5% damping response spectrum, i.e., 1.76 g in the longitudinal direction

(Figure 2). However, this value is so high that the free-surface would make an angle of 60

(tan-11.76) with the horizontal [7] and expose the inlet pipes (quenchers) during SSE level

shaking. Similarly, a conservative value of the base shear would be 1.769.81951,000/106 =

16.46 MN. Again, this value is so high that a single fixed-support will not be able to resist the

entire seismic force in the longitudinal direction of the tank. Therefore, an approximate

conservative analysis was not practical for this tank and it was decided to determine the sloshing

wave height and the base shear from the fluid-structure interaction analysis. As a first step in this

analysis, spectrum-compatible histories of floor motion were generated as described next.



Simulated Histories of Floor Motion

One set of tri-axial floor motions was simulated for the dynamic analysis of the VVPSS tank by

using the following steps:

Three components of a natural (recorded) earthquake motion were selected.

Fourier amplitude and phase spectra of three components were generated from the Fast

Fourier Transform (FFT).

The Fourier amplitudes were simultaneously modified in all three directions such that the

response spectra of the modified motions match the 3% damping response spectra shown in

Figures 2, 3 and 4. The phase spectra in three directions were not modified; the original

phase spectra were preserved in all three directions.

The simulated floor motions were corrected such that the integration and double-integration

of the acceleration histories produces velocity and displacement histories without any trends.

Figure 5 shows the acceleration, velocity and displacement histories of the simulated floor

motion in the longitudinal direction of the tank. As expected from the shape of the response

spectrum in the longitudinal direction (Figure 2), the simulated floor motion is dominated by the

0.5 Hz frequency. Figure 6 compares the 3% damping response spectrum of the simulated floor

motion with the 3% damping floor response spectrum in the longitudinal direction. The match

between the target and the achieved spectra is quite good. Similar matches between the target

and the achieved response spectra were also achieved in the other two directions.

Figure 5. Acceleration, velocity and displacement histories of simulated floor motion in the

longitudinal direction of the tank.



Figure 6. Comparison between the 3% damping response spectrum of simulated floor motion

in longitudinal direction (Figure 5) and the 3% damping SSE level envelope

response spectrum in the same direction (Figure 2).

Figure 7. Acceleration, velocity and displacement histories of the simulated floor motion in

the transverse direction of the tank.



Figure 8. Acceleration, velocity and displacement histories of simulated floor motion in the

vertical direction of the tank.

Figures 7 and 8 show the simulated floor motions in the transverse and vertical directions of the

tank. The motion in the transverse direction has some high-frequency content consistent with the

shape of the response spectrum in that direction (Figure 3). The motion in the vertical direction is

dominated by high frequencies, consistent with the shape of the response spectrum in the vertical

direction (Figure 4). Only one set of tri-axial floor motions was generated to confirm the

preliminary design of the VVPSS tank. The final design of the tank should be checked with at

least 3 sets of tri-axial floor motions.



Convective Modes of Vibration

Since convective (sloshing) modes of vibration have low natural frequencies, they are not

affected by the deformation of the tank shell. Therefore, the convective response can be

computed by assuming that the tank wall is rigid. A finite element model of the tank shell and

fluid system was created in ANSYS [8] as shown in Figure 9. In this model, the tank shell was

assumed rigid and the saddle supports were not included. The water in the tank was explicitly

modeled by ANSYS [8] fluid-element Fluid 80. The convective modes of vibration were

extracted from the eigen-value analysis of the tank-liquid system. Table 1 lists the fundamental

frequencies of the convective modes in the longitudinal and transverse directions of the tank.

Figure 9. Finite element model of the tank shell and fluid system for extracting convective

modes of vibration.

The sloshing frequencies in the transverse direction of the tank were confirmed with published

literature. According to Budiansky [9], the sloshing frequencies in a circular canal are given by

the following equation:

/

(2)

in which, g = acceleration due to gravity; R = radius of the circular canal. i values are provided

for different water levels in the canal [9]. Corresponding to the water level in the VVPSS tank,



1 = 1.41 and 2 = 4.77. Substituting the values of various variables in Equation 2 gives f1 =

0.35 Hz and f2 = 0.64 Hz. These values agree well with those obtained from the ANSYS [8]

analysis (listed in Table 1).

There are no analytical studies available in the longitudinal direction of cylindrical horizontal

tanks. Patkas et al. [10] have proposed the use of equivalent rectangular container for which the

dynamic characteristics can be obtained using the study by Abramson [11]. The natural

frequencies obtained by the use of equivalent rectangular container are 0.053 Hz for 1st mode and

0.153 Hz for the 2nd mode. These agree well with those obtained from the ANSYS [8] analysis

(Table 1).

Table 1. Fundamental natural frequencies and periods of sloshing modes of vibration.

Direction Mode Frequency

(Hz) Period

(s)

Longitudinal 1st 0.0562 17.8

Longitudinal 2nd 0.163 6.12

Longitudinal 3rd 0.268 3.73

Longitudinal 4th 0.372 2.69



Transverse 1st 0.358 2.79

Transverse 2nd 0.627 1.60

Table 2 lists the cumulative masses associated with the convective modes of vibration in the

longitudinal and transverse directions of the tank. These masses were obtained by adding the

mass participations of the first few modes of vibration. Hence, these are lower-bound values of

the convective masses. The remaining liquid mass was considered impulsive, leading to a

somewhat conservative estimate of the foundation forces. The mass of the liquid in the tank is

675,000 kg. In the longitudinal direction, the convective liquid mass is 72% of the total liquid

mass. In the transverse direction, the convective liquid mass is only 42% of the total liquid mass.

The convective liquid mass is smaller in the transverse direction because the tank wall constrains

the liquid more in the transverse direction than in the longitudinal direction. For similar water



level, Karamanos et al. [12] report a higher value of the convective mass (about 54% of the total

liquid mass) in the transverse direction. For the longitudinal direction, the equivalent rectangular

container proposed by Patkas et al. [10] predicts 20% higher convective mass than that obtained

from ANSYS [8] analysis. This discrepancy in convective masses can be studied in more detail

during the final design. In this study of preliminary-design a lower value of the convective mass

based on ANSYS [8] results was used to obtain conservative estimates of foundation forces.

Table 2. Cumulative convective liquid masses in longitudinal and transverse directions of the

tank.

Direction Convective Liquid Mass Percent of Total Liquid Mass

Longitudinal 483,660 kg 72%

Transverse 281,500 kg 42%



Impulsive Modes of Vibration

The impulsive liquid mass was obtained by subtracting the convective liquid mass (Table 2)

from the total liquid mass of 675,000 kg. In the vertical direction, there is no sloshing response,

therefore, the entire liquid mass is impulsive. Table 3 lists the impulsive liquid masses in the

longitudinal, transverse and vertical directions of the tank.

Table 3. Impulsive liquid masses in the transverse, longitudinal and vertical directions of the

tank.

Direction Impulsive Liquid Mass Percent of Total Liquid Mass

Longitudinal 191,340 kg 28%

Transverse 393,500 kg 58%

Vertical 675,000 kg 100%

To extract the impulsive modes of vibration of the tank, a full model of the tank was constructed

in ANSYS [8] as shown in Figure 10. In the longitudinal direction, the portions of the concave

end walls that come in contact with the water were made denser such that their mass increases by

the mass of the impulsive liquid in the longitudinal direction. In the transverse direction, the tank

shell which comes in contact with the liquid was made denser such that the mass of the tank shell

increases by the mass of the impulsive liquid in the transverse direction. In the vertical direction,

the tank shell which comes in contact with the liquid was made denser such that the mass of the

tank shell increases by the mass of the impulsive liquid in the vertical direction. Two support

conditions were analyzed in the longitudinal direction of the tank:

1. All fixed supports; and

2. Only one (middle) fixed support and other sliding supports to allow free thermal expansion

of the tank.

The modes of vibration were extracted from the eigen-value analysis of the tank. Table 4 lists the

natural frequencies and periods of the impulsive modes of vibration in the longitudinal,

transverse and vertical directions of the tank. Note that when the number of fixed supports is

reduced from 9 to 1 in the longitudinal direction the natural frequency reduces from 10.96 Hz to

3.76 Hz — a reduction by a factor of about 3. This implies that the flexibility in the longitudinal

direction of the tank is due to the supports rather than the shell. Note also that the impulsive



frequencies are much higher than the convective frequencies reported in Table 1. This further

justifies uncoupling of the convective response from the impulsive response.

Figure 10. Finite element model of the tank for extracting impulsive modes of vibration.

Table 4. Impulsive frequencies and periods in the transverse, longitudinal and vertical

directions of the tank.

Direction Impulsive Frequency Impulsive Period

Longitudinal — All fixed supports 10.96 Hz 0.0913 s

Longitudinal — Only 1 fixed support 3.76 Hz 0.266 s

Transverse 5.71 Hz 0.175 s

Vertical 43.91 Hz 0.0228 s



Sloshing Response

For sloshing response water in the tank was modeled explicitly by ANSYS [8] fluid-elements.

The time history analysis of sloshing response was performed by using all convective modes of

vibration. The damping for the sloshing modes of vibration was assumed to be 0.5% of critical as

per ITER Guidelines [6]. Figure 11 shows the history of sloshing wave height (vertical motion of

free-surface) due to shaking in the longitudinal direction of the tank. The maximum sloshing

wave height is 38.5 cm. Since the inlet pipes are immersed 2.69 m into water, they cannot be

exposed by SSE level seismic shaking. Frequency analysis of the sloshing response showed that

the sloshing wave height in the longitudinal direction was controlled by the 6th convective mode

of vibration. Figure 12 shows the history of sloshing wave height (vertical motion of free-

surface) in the transverse direction of the tank. The maximum sloshing wave height in the

transverse direction is 137 cm. Due to low damping (0.5% of critical) the sloshing response is

significant even after the seismic shaking has stopped at about 50 s (Figures 5, 11 and 12).

Figure 11. Sloshing wave height in the longitudinal direction of the tank.

Figure 12. Sloshing wave height in the transverse direction of the tank.



Base Shear and Reactions

The base shears were computed from the time history analysis using convective modes listed in

Table 1 and all impulsive modes of vibration of tank-liquid system. Table 5 lists the natural

frequencies and masses of a few convective and impulsive modes of vibration. The impulsive

masses in Table 5 are higher than those in Table 3 because the masses in Table 3 are for the

liquid only. The masses in Table 5 are for the entire tank-liquid system, which includes the

liquid, tank shell and the supports. The last column in Table 5 lists the response accelerations

read from the response spectra shown in Figures 2 and 3. In the longitudinal direction the

convective mass is 1.7 times the convective mass in the transverse direction. But the low

convective frequency in the longitudinal direction causes low convective accelerations hence low

convective base shear of only 0.0729 MN. In the transverse direction the convective base shear is

1.53 MN which is 33 times the value in the longitudinal direction.

Table 5. Natural frequencies, masses and response accelerations of a few modes of vibration of

tank-liquid system.

Model Type Mode No. Frequency

(Hz) Mass (kg)

Acceleration (g)

Convective, longitudinal

1 0.0562 410,925 0.00265

2 0.163 43,596 0.0307

3 0.268 14,219 0.131

Convective, transverse 1 0.358 273,312 0.474

2 0.627 7,651 1.754

Impulsive, longitudinal, all fixed

1 11 388,380 0.199

2 19.7 22,690 0.142

3 26.6 6,780 0.134

Impulsive, longitudinal, middle fixed 1 3.76 452,000 0.153

Impulsive, transverse

1 5.71 122,800 0.696

2 10 60,930 0.820

3 20.8 21,170 0.212

Since the tank is required to remain operational during the SSE level shaking, material yielding

was not allowed. Hence, material nonlinearities were not modeled in the analysis. The response

history analysis of impulsive response was performed by using all impulsive modes of vibration.

The peak impulsive base shear in the longitudinal direction of the tank is 0.96 MN when all

supports are fixed in the longitudinal direction. When only the middle support is fixed the



impulsive base shear decreases to 0.888 MN. This is because a reduction in the natural frequency

from 10.96 Hz to 3.76 Hz causes a reduction in the impulsive response acceleration according to

Figure 2 which then reduces the impulsive base shear.

Table 6 provides a summary of impulsive and convective base shears in the horizontal directions

and impulsive reaction in the vertical direction of the tank. These values are somewhat higher

than those obtained by multiplying the last two columns in Table 5 because: (a) only first few

modes are listed in Table 5, and (b) the spectra of simulated motions are somewhat different

from the target spectra. The peak impulsive and convective base shears were combined by using

the SRSS (square-root-of-sum-of-squares) method. The net peak base shears and vertical

reaction in three directions are listed in the last column of Table 6.

Table 6. Summary of impulsive, convective and net base shears in three directions of the tank.

Direction Convective Impulsive Net

Longitudinal — All fixed supports 0.0729 MN 0.96 MN 0.963 MN

Longitudinal — Only 1 fixed support 0.0729 MN 0.888 MN 0.891 MN

Transverse 1.53 MN 2.22 MN 2.7 MN

Vertical NA 9.9 MN 9.9 MN



Conclusions

An approximate conservative seismic analysis of a very large horizontal water storage tank

provided impractical results of liquid sloshing and support reactions. These would have required

very expensive mitigation measures. A detailed finite element analysis was performed to obtain

the following results:

1. Only about 28% of the liquid moves with and deforms the tank wall in the longitudinal

direction. The remaining 72% of the liquid sloshes independently of the tank wall and

generates negligible seismic forces.

2. The sloshing wave height in the longitudinal direction of the tank is only about 12% of the

liquid height because the first few sloshing modes are not excited by seismic shaking due to

their very low natural frequencies.

3. The net base shear in the longitudinal direction of the tank is only about 10% of the total

weight of the liquid-filled tank although the peak spectral acceleration in the longitudinal

direction is about 1.76 g.

4. Due to the small sloshing wave height, baffles are not needed to control sloshing in the tank.

5. Only middle support of the tank can be fixed in the longitudinal direction while other

supports can be on low-friction sliding bearings to accommodate thermal expansion of the

tank.

When the tank is very large and/or the seismic input motion is dominated by a few frequencies,

an approximate conservative solution may not be practical. A detailed finite element analysis can

be performed to accurately calculate liquid sloshing and support forces. Results of detailed

analysis should always be checked with simpler analysis.



Disclaimer

The views and opinions expressed herein do not necessarily reflect those of the ITER

Organization. The results should be considered preliminary until they are accepted by the ITER

licensing agency.



Acknowledgments

This paper is based on a project funded by ITER organization. The support and feedback of the

project review committee is greatly appreciated. ITER organization is thanked for the permission

to publish this paper. Two anonymous reviewers of the manuscript provided a thorough review

and several helpful suggestions.



References

[1] Housner, G. W. The dynamic behavior of water tanks. Bulletin of the Seismological

Society of America, 53(2), 1963, 381–387.

[2] Haroun, M. A., and Housner, G. W. Seismic design of liquid-storage tanks. Journal of

Technical Councils, ASCE, New York, 107(1), 1981, 191–207. ASCE, New York, 1984,

255–370.

[3] Veletsos, A. S. Seismic response and design of liquid storage tanks. Guidelines for the

Seismic Design of Oil and Gas Pipeline Systems, ASCE, New York, 1984, 255–370.

[4] Malhotra, P., Wenk, T., and Wieland, M. Simple Procedure for Seismic Analysis of

Liquid-Storage Tanks. J. Struct. Eng. International, IABSE, 10(3), 2000, 197-201.

[5] Engage. Tender design – Tokamak Complex PBS 62.11, PBS 62.14 and PBS 62.74

Generation of the floor response spectra. ENG-40-CR-110001-CW, 2011.

[6] ITER. Load Guidelines. Version 6.0, 20 April 2012.

[7] Malhotra, P. K. Earthquake Induced Sloshing in Cone and Dome Roof Tanks with

Insufficient Freeboard. J. Struct. Eng. International, IABSE, 16(3), August 2006, 222-

225.

[8] ANSYS Inc. ANSYS Multiphysics 13.0, Southpointe, 275 Technology Drive, Canonsburg,

PA 15317, 2012.

[9] Budiansky, B. Sloshing of liquids in circular canals and spherical tanks. Journal of

Aerospace Science, 27, 1960, 161-173.

[10] Patkas, L. A., Karamanos, S. A., and Platyrrachos, M. A. Seismic Analysis and Design of

Horizontal-Cylindrical Industrial Vessels. Proceedings of the 4th European Workshop on

the Seismic Behaviour of Irregular and Complex Structures, Thessaloniki, Greece, Paper

No. 22, 2005, August.

[11] Abramson, H. N. The Dynamic Behavior of Liquids in Moving Containers, Southwest

Research Institute, NASA SP-106, 1966, Washington, DC.

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Design of Horizontal-Cylindrical and Spherical Industrial Vessels, J. Pressure Vessel

Technology, ASME, 128, August 2006, 328-340.



Nomenclature

a = Acceleration

cm = Centimeter

d = Displacement

F = Natural frequency in Hz

g = Acceleration due to gravity (9.81 m/s2)

ITER = International Thermonuclear Experimental Reactor

m = Meter

mm = Millimeter

MN = Mega Newton (106 Newton)

MRP = Mean return period

PD = Peak deformation (also known as spectral deformation; sometimes called

spectral displacement)

PPA = Peak pseudo-acceleration (also known as spectral acceleration)

PPV = Peak pseudo-velocity (also known as spectral velocity)

s = Seconds

SSE = Safe Shutdown Earthquake

v = Velocity

VVPSS = Vacuum Vessel Pressure Suppression System

Seismic Sloshing in a Horizontal Liquid-Storage Tank

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