Keywords: metallic tanks, seismic analysis, finite elements, stress and displacement envelopes, hydrodynamic pressure.

During an earthquake, the dynamic fluid pressures developed on the liquid storage tank thin-walls and foundations are of great importance in the tank performance and response, as well as in their seismic resistant design. They may cause two major tank shell structural instabilities: elephant-foot bulge type of buckling and diamond buckling by shell-crippling.

Some recent experience on the seismic analysis and design of metallic tanks by different approaches, through an R&D project entitled Seismic Response of Tanks by the Finite Element Method (FEM), permitted:

• Review of rigid tank method complete formulation with all modes;
• Development of tank shell equations by the so-called Sanders theory, as mentioned by Haroun [1] and applied by Barros [2];
• Development of time history response results of the fluid structure interaction (FSI) between tank contained liquid and the FEM model of tank shell, namely for hydrodynamic pressure, tank stress resultants, generalized displacements, surface elevation, base shears and base moments, as presented by Barros [2,3], and permitting the evaluation of response envelopes;
• Development of a FEM formulation to model FSI between liquid and tank by FEM model of tank shell and roof, as well as a FEM model of contained liquid as a degenerated solid, as explained by Barros and Alves [4];

Seismically excited thin metallic tanks, bottom supported, are modelled by Sanders shell theory. The complete boundary value problem is solved taking into account the fluid-structure interaction, since the hydrodynamic actions on the shell tank wall depend on the response of the shell and liquid contents to the seismic motions of the foundation soil. Assuming small liquid displacements. A rigorous mathematical treatment involves deriving an expression for the liquid velocity potential satisfying Laplace equation and appropriate boundary conditions at the liquid surface and at the shell-liquid interface.

The hydrodynamic pressures along tank wall include both impulsive and convective contributions, and are expressed by an infinite series that can be conveniently truncated after a few terms, depending on the degree of desired accuracy. The tank shell is also discretized in a finite number of ring elements. An interactive computer package was developed in Fortran language, calibrated with other tank results and conveniently tested and optimised for design office use [2,3].

In order to determine envelopes for the hydrodynamic pressures on a given storage steel tank, under type 1 and 2 seismic actions of Portuguese design standards, a number of artificial earthquakes were generated [5] to test the convergence of some typical seismic response variables. It was found out that 10 to 14 earthquakes are required, for converged responses of tanks (under a tight relative error control criteria), with which hydrodynamic pressure envelopes were then evaluated.

Also, the analyses permit to ascertain the safety of the tank's content with regard to possible cavitations of the stored liquids, when excited by earthquakes satisfying two distinct seismic code provisions. Additionally, the methodologies permit to visualize the free surface effects on the stored liquid during the earthquake occurrence, and therefore evaluate the value of freeboard required to prevent spilling of tank's content (if uncapped) or to prevent roof hydrodynamic overpressures that might cause roof collapse.

Response envelopes of generalized displacements and stresses were determined, which reveal very useful in ascertaining tank design under the strength stiffness and stability viewpoints. Particularly, the envelope of axial stresses permits to assess the onset of elephant-foot bulge type of buckling.

1

Haroun, M.A., "Dynamic Analyses of Liquid Storage Tanks", Report to the National Science Foundation, EERL 80-04, California Institute of Technology, Pasadena, 1980.

2

Barros, R.C., "Modelação da Resposta Sísmica de Tanques Circulares pelo M.E.F., incluindo Interacção Tanque-Líquido", in "Proceedings of the VI Congresso Nacional de Mecânica Aplicada e Computacional", Paulo Vila Real and José J. Grácio (Editors), Vol. 2, Tema: Mecânica dos Sólidos, pp. 1203-1212, A.P.M.T.A.C., Lisboa, 2000, (in Portuguese).

3

Barros, R.C., "Some Developments on Vibration Control for Tank Shells and Pipelines", Oral presentation on 25th September 2001 in the Engineering Meeting on Intelligent Structures, EMIS 2001, Ischia, Italy, 2001.

4

Barros, R.C., and Alves, R.F., "Seismic Response of Thin Shell Storage Tanks", in "Earthquake Resistant Engineering Structures III", C.A. Brebbia & A. Corz (Editors), Section 9: Special Structures, pp. 733-743, WIT Press, Southampton, U.K., 2001.

5

Vanmarcke, E.H., Fenton, G.A., and Heredia-Zavoni, E., "SIMQKE-II: Conditioned Earthquake Ground Motion Simulator", User's Manual, Version 2.1, Princeton University, Princeton, New Jersey, 1999.

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