Seismic Sloshing in a Horizontal Liquid-Storage Tank

Transcription

1 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 5 th 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 StrongMotions Inc. (781) April 9, 2014 Page 1 of 23

2 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 predesign 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. StrongMotions Inc. (781) April 9, 2014 Page 2 of 23

3 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 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 liquidfilled 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/m 3. Figure 1. Longitudinal section of the VVPSS tank at ITER facility (all dimensions are in millimeters). StrongMotions Inc. (781) April 9, 2014 Page 3 of 23

4 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/s 2. 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. StrongMotions Inc. (781) April 9, 2014 Page 4 of 23

5 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 StrongMotions Inc. (781) April 9, 2014 Page 5 of 23

6 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. StrongMotions Inc. (781) April 9, 2014 Page 6 of 23

7 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 ) with the horizontal [7] and expose the inlet pipes (quenchers) during SSE level shaking. Similarly, a conservative value of the base shear would be ,000/10 6 = 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. StrongMotions Inc. (781) April 9, 2014 Page 7 of 23

8 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. StrongMotions Inc. (781) April 9, 2014 Page 8 of 23

9 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. StrongMotions Inc. (781) April 9, 2014 Page 9 of 23

10 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. StrongMotions Inc. (781) April 9, 2014 Page 10 of 23

11 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, StrongMotions Inc. (781) April 9, 2014 Page 11 of 23

12 1 = 1.41 and 2 = Substituting the values of various variables in Equation 2 gives f 1 = 0.35 Hz and f 2 = 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 Hz for 1 st mode and Hz for the 2 nd 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 1 st Longitudinal 2 nd Longitudinal 3 rd Longitudinal 4 th Longitudinal 5 th Longitudinal 6 th Transverse 1 st Transverse 2 nd 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 StrongMotions Inc. (781) April 9, 2014 Page 12 of 23

13 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% StrongMotions Inc. (781) April 9, 2014 Page 13 of 23

14 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 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 StrongMotions Inc. (781) April 9, 2014 Page 14 of 23

15 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 Hz s Longitudinal Only 1 fixed support 3.76 Hz s Transverse 5.71 Hz s Vertical Hz s StrongMotions Inc. (781) April 9, 2014 Page 15 of 23

16 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 6 th convective mode of vibration. Figure 12 shows the history of sloshing wave height (vertical motion of freesurface) 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. StrongMotions Inc. (781) April 9, 2014 Page 16 of 23

17 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 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 Mass Acceleration (Hz) (kg) (g) , Convective, longitudinal , , Convective, transverse , , , Impulsive, longitudinal, all fixed , , Impulsive, longitudinal, middle fixed , , Impulsive, transverse , , 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 StrongMotions Inc. (781) April 9, 2014 Page 17 of 23

18 impulsive base shear decreases to MN. This is because a reduction in the natural frequency from 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 MN 0.96 MN MN Longitudinal Only 1 fixed support MN MN MN Transverse 1.53 MN 2.22 MN 2.7 MN Vertical NA 9.9 MN 9.9 MN StrongMotions Inc. (781) April 9, 2014 Page 18 of 23

19 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. StrongMotions Inc. (781) April 9, 2014 Page 19 of 23

20 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. StrongMotions Inc. (781) April 9, 2014 Page 20 of 23

21 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. StrongMotions Inc. (781) April 9, 2014 Page 21 of 23

22 References [1] Housner, G. W. The dynamic behavior of water tanks. Bulletin of the Seismological Society of America, 53(2), 1963, [2] Haroun, M. A., and Housner, G. W. Seismic design of liquid-storage tanks. Journal of Technical Councils, ASCE, New York, 107(1), 1981, ASCE, New York, 1984, [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, [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, [5] Engage. Tender design Tokamak Complex PBS 62.11, PBS and PBS Generation of the floor response spectra. ENG-40-CR CW, [6] ITER. Load Guidelines. Version 6.0, 20 April [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, [8] ANSYS Inc. ANSYS Multiphysics 13.0, Southpointe, 275 Technology Drive, Canonsburg, PA 15317, [9] Budiansky, B. Sloshing of liquids in circular canals and spherical tanks. Journal of Aerospace Science, 27, 1960, [10] Patkas, L. A., Karamanos, S. A., and Platyrrachos, M. A. Seismic Analysis and Design of Horizontal-Cylindrical Industrial Vessels. Proceedings of the 4 th 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. [12] Karamanos, S. A, Patkas, L. A., and Platyrrachos, M. A. Sloshing Effects on the Seismic Design of Horizontal-Cylindrical and Spherical Industrial Vessels, J. Pressure Vessel Technology, ASME, 128, August 2006, StrongMotions Inc. (781) April 9, 2014 Page 22 of 23

23 Nomenclature a = Acceleration cm = Centimeter d = Displacement F = Natural frequency in Hz g = Acceleration due to gravity (9.81 m/s 2 ) ITER = International Thermonuclear Experimental Reactor m = Meter mm = Millimeter MN = Mega Newton (10 6 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 StrongMotions Inc. (781) April 9, 2014 Page 23 of 23