Implementation of Fourier Series in the Seismic Response Spectrum Analysis of Ground Supported Tanks

This paper describes the finite element considerations employed in a seismic response spectrum analysis of a ground supported, liquid containing tank. Like many axisymmetric cylindrical vessels, the gross seismic response to an input response spectrum can be categorized by a simplified lumped mass model that includes both the mass of the tank proper in combination with the associated mass of multiple fluid levels. This simplified response may be utilized to determine the initial sizing of the supporting configuration, but lacks the ability to properly address the fluid-structure interaction that creates sloshing loads on the tank walls. The most obvious method to address the fluid-structure interaction when considering the finite element method is to build a three-dimensional model of the tank, including, but not limited to the shell courses, the top and bottom heads (for a vertical vessel), and any tank supports. The inclusion of the fluid effects may now be incorporated with “contained fluid” finite elements, however, for tanks of any significant volume, the number of finite elements can easily exceed 100,000 and the number of degrees of freedom can sore from as few as 300,000 to as many as 500,000 or more. While these types of finite element analysis problems can be solved with today’s computer hardware and software, it is not desirable in any analysis to have that volume of information that has to be subjected to the nuclear QA environment (if at all possible). With these items in mind, the methodology described in this paper seeks to minimize the number of degrees of freedom associated with a response spectrum analysis of a liquid filled, vertical cylindrical tank. The input response spectra are almost always provided in Cartesian coordinates, while many, if not most liquid containing pressure tanks are almost always axisymmetric in geometry without having benefit of being subjected to an axisymmetric load (acceleration in this case) due to the specified seismic event. The use of harmonic finite elements for both the tank structure and the contained fluid medium permit the efficiencies associated with an axisymmetric geometry to be leveraged when the seismic response spectrum is formulated in terms of a Fourier series and combined to regain the effects of the two orthogonal, horizontally applied accelerations as a function of frequency. The end result as discussed and shown in this paper is a finite element model that permits a dense mesh of both the fluid and the structure, while economizing on the number of simultaneous equations required to be solved during the chosen finite element analysis.