Becht Nuclear Services recently applied ANSYS® multi-surface, with fluid-structure interaction (FSI) capabilities to evaluate the effects of seismic loading on vessel internals that are immersed in the fluid. Before proceeding with the analysis of the actual vessel, the FSI capabilities were benchmarked to an analytical solution of a simpler liquid-filled flexible tank under seismic loading.
A fixed-base, flat-bottom, cylindrical tank, with a height/radius ratio (H/R) similar to the vessel to be analyzed, is modeled in ANSYS® as shown in Figure 1. The tank is filled with an inviscid liquid that is free to slosh, without impacting a roof. SOLID186 (3-D 20-node) structural solid elements are used for the vessel. The use of shell structural elements is also investigated and the performance compared with solid elements. FLUID220 (3-D 20-node) acoustic elements are used to simulate the fluid. A characteristic of the acoustic equations is that there is no net flow and the fluid response is limited to small displacements.
The tank model is subjected to horizontal shaking only, and a response spectrum analysis is performed to determine the pressure profile on the tank wall. A broad-banded spectrum anchored at 0.25g peak ground acceleration is used as input. The input spectrum is shown in Figure 2.
There are a number of closed-form analytical solutions available with various simplifying assumptions to estimate the response of liquid-filled structures under seismic loading. Among them the Housner (1954) approximate method is widely used. An accurate solution depends on various parameters including the flexibility of the structure. For cylindrical tanks the parameters of interest include the tank height/radius ratio (H/R) and the liquid height/radius (Hl/R). The dynamic pressure on the wall under seismic loading consists of convective (sloshing) and impulsive mode contributions. The impulsive contribution is affected significantly by the flexibility of the tank wall, whereas the convective contribution is insensitive to this flexibility and may be evaluated considering the tank wall to be rigid.
In Brookhaven National Laboratory report BNL 52361 (1995) tabular results, based on analytical solutions for rigid tanks, are given for relatively broad tanks (H/R ≤ 1) as obtained from Veletsos (1984) which considered H/R values from 0.3 to 5.0 for rigid tanks. The maximum dynamic pressure on the wall of a rigid tank occurs at the bottom of the tank. The maximum hydrodynamic effect for broad flexible tanks can be approximately obtained from the relevant expressions for a rigid tank solution by simply replacing the maximum ground acceleration in these expressions by the spectral value of the pseudo-acceleration corresponding to the fundamental natural frequency of the tank-liquid system. Dunkerley’s approximate method, as reported in BNL 52361, can be used to obtain the tank-liquid fundamental natural frequency. However, for more slender flexible tanks (H/R > 1) of interest herein with H/R = 1.8, this approximate approach is insufficient. In this case, more accurate results from Yang (1976), Tang and Chang (1993) and Padmanaban (1992 and 1996) are used to obtain the analytical solution presented graphically in Figure 3. In this case of a more slender flexible tank, the maximum dynamic pressure occurs not at the tank bottom, but instead more than halfway up the tank wall.
The results from the finite element model are found to compare well with the corresponding analytical solution. The first sloshing mode of the fluid was determined to be 0.25 Hz in both the finite element model and the analytical solution. The resulting pressure profile is compared to the analytical solution in Figure 3. For the analytical solution the impulsive, convective, and resulting total pressure profiles of the fluid on the tank wall are shown in Figure 3. The finite element model results are a combination of the impulsive and convective modes, and results are shown for both a shell and solid structural model of the tank. In both the analytical and the finite element model response spectrum analysis the total dynamic pressure profile is obtained using the square root of sum of the squares (SRSS) modal combination procedure.
The finite element model results and the analytical solution are in good agreement, and the use of solid structural elements provides slightly better results than shell structural elements. The use of ANSYS® FSI capabilities provides greater flexibility in modeling complex geometric structures with contained fluid than the limited available analytical solutions. If the expected fluid response is not limited to small displacements, or slosh impacting with a roof structure can occur, then the full nonlinear Navier-Stokes fluid equations may need to be used, which would require techniques other than the use of acoustic elements.
ANSYS, 2015, ANSYS Mechanical APDL Acoustic Analysis Guide, Release 16.2, ANSYS, Inc., Canonsburg, Pennsylvania.
BNL 52361, 1995, Seismic Design and Evaluation Guidelines for the Department of Energy High-Level Waste Storage Tanks and Appurtenances, K. Bandyopadhyay, et al., Brookhaven National Laboratory, Upton, New York.
Housner, G. W., 1954, Earthquake Pressures on Fluid Containers, California Institute of Technology, Pasadena, California.
Padmanaban, S., 1992, Dynamic Response of Liquid-Waste Storage Tanks Subjected to Coherent and Incoherent Ground Motion, Master of Science Thesis, Rice University, Huston, Texas.
Padmanaban, S., 1996, Dynamic Response of Hazardous Liquid-Waste Storage Tanks used in Nuclear Facilities, PhD Thesis, Rice University, Huston, Texas.
Tang, Y., and Y. W. Chang, 1993, Free Vibration Analysis of Partially Filled Liquid Storage Tanks, ANL/RE-93/3, Argonne National Laboratory, Argonne, Illinois.
Veletsos, A. S., 1984, “Seismic Response and Design of Liquid Storage Tanks,” in Guidelines for the Seismic Design of Oil and Gas pipeline Systems, American Society of Civil Engineers, New York, New York.
Yang, J. Y., 1976, Dynamic Behavior of Fluid-Tank Systems, Rice University, PhD Thesis, Civil Engineering Department.
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