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## Numerical Analyses

The numerical techniques, specially the finite element method, are powerful tools to analyze the problem of unanchored tanks because of their flexibility and ability to model all the complications involved with the analysis of such tanks. Many researchers have employed these techniques to analyze the problem. Auli, Fischer and Rammerstorfer [11] presented an analysis for uplifting of unanchored tanks in 1985. They used the finite element technique to solve an axisymmetric uplift problem whereas the bottom plate experiences a uniform uplift all around circumference. As an approximation, they considered the tank as a cylindrical shell resting on nonlinear springs without a bottom plate. Hence, the number of degrees of freedom and the number of elements were reduced, and no contact elements were needed to model the bottom plate. The stiffness of the uplifted bottom plate was represented by equivalent Winkler springs distributed on the lower edge of the shell wall. In order to determine the nonlinear characteristics of these springs, the bottom plate was modeled as a strip resting on a rigid foundation, and loaded by uniformly distributed as well as uplift forces. In addition to these forces, two springs representing bending and extensional stiffness of the tank wall were introduced at the free end, assuming that the fluid pressure acted only on the bottom plate and not on the tank shell. Based on resulting relationship between the uplifting force and the uplifting displacement, the vertical restraining action due to the weight of the fluid resting on the uplifted portion of the base plate was obtained and applied around the circumference of the shell. In Wozniak and Mitchell's model [292], this vertical resistance force was equal to the maximum hold down force, and was applied around the entire circumference except in the contact region leading to an overestimation of the resistance of the bottom plate, whereas in Auli's model this force was a deflection dependent force varied around the circumference.

Using a general purpose finite element computer code, Barton and Parker [16] investigated the seismic response of the liquid-filled cylindrical storage tanks. A time history analysis of a three-dimensional finite element model of an unanchored tank and its contents with a specified gap conditions between the tank base and the supporting floor to allow lift-off of the base was performed. Neither material nor geometric nonlinearities were considered in the analysis and the tank was assumed to be subjected to only one direction of horizontal excitation. Furthermore, the fluid hydrodynamic problem was eliminated by lumping assumed percentage of the fluid mass to the tank wall and base plate. Results of the investigation indicated that stresses in the tank and resultant loads on the floor of an unanchored tank were much greater than those for a rigidly restrained tank and showed the importance of carefully considering the restraint conditions when performing seismic design calculations on storage tanks.

In 1986, a simplified method of analysis of unanchored tanks under static lateral loads was developed by Peek [222]. He used a similar analytical concept to that developed earlier by Auli et al. The interaction between the shell and the bottom plate was modeled by placing axial springs all around the lower edge of the shell. Axisymmetric solution of the bottom plate subjected to a uniform amount of uplift all around the circumference was used in order to obtain a force-deflection relationship for equivalent Winkler springs which represent the restraining action of the base plate. Knowing the relationship between the hold-down force and the uplift displacement, and using axisymmetric shell elements, the analysis of cylindrical shell was performed based on linear shell theory. It was also assumed that both the base plate and the shell remain elastic, and foundation is rigid. In a following work, Peek [219] used the finite difference energy method in conjunction with a Fourier series representation of the circumferential variation of displacements and obtained a numerical solution of two dimensional contact problem of a base plate under static lateral loads. Nonlinearities due to contact finite displacements and yield of the steel were included in the analysis. The equations for the shell were linearized to uncouple the equations for the Fourier displacement coefficients in the cylindrical shell and to enable the degrees of freedom for the shell to be eliminated by means of static the condensation at a little computational cost. He compared the analytical results to experimental results and reported that they were in good agreement in some cases and not so good in others. He discussed a number of effects that could give rise to such differences and stated that in most cases they represent experimental conditions that were not known or modeled in the analysis. The analysis results were also compared to those from a simplified analysis in which the hold down action of the base plate was modeled by means of nonlinear Winkler springs. Peek concluded that large membrane stresses developed in the bottom plate carry most of the load on the uplifted region. In addition, Peek and Jennings [221] suggested that in order to increase the capacity of an unanchored tank to withstand lateral loads due to tilting, tank wall could be pre-uplifted all around the circumference by means of a ring filler. Depending on the frequencies of earthquake, lateral loads generated by ground motion for a preuplifted tank could be higher or lower than those for a tank without preuplift. However, they recommended that further investigations are needed to study the effectiveness of the method and to investigate the behavior of the base plate of a pre-uplifted tank before and after the earthquake.

In 1988, Haroun and Badawi [12] modeled the base plate in both its strip and circular configurations and investigated its nonlinear behavior under equivalent static uplifting forces using an approximate energy-based approach. This analysis differs from other available analyses in that the plate is modeled as a circular plate with an uplifted, crescent-shaped region rather than being modeled as a strip. The behavior of the plate under both small and large deflection assumptions, and the effects of stretching of its midplane were studied. The reliability of the crescent-shaped model was confirmed through a comparison with the analysis of an axisymmetric circular model. The governing set of nonlinear equations was derived for the system by minimizing its potential energy with respect to the generalized coordinates. Upon solving these equations, the bending moments and stresses were evaluated throughout the plate by making use of the generalized coordinates. Results of these models clearly showed that for moderate values of the ground acceleration, the membrane actions developed in the plate increased its load-carrying capacity, yielding a much lower value for uplift displacements. They noted that the analysis under axisymmetric conditions yielded results comparable to those of the asymmetric case, but the former analysis is much simpler; it showed that the assumed deflection shape of the plate is an important factor only under small deflection assumptions since the solution is governed by the bending actions. Under large deflections, the membrane actions dominate and loads are carried mainly by inplane forces. A concurrent work, also was performed by Haroun and Bains [13], sought the same characteristics of the base plate by a nonlinear finite element shell program to study the static behavior of uplifted unanchored tanks. The program was extended to analyze the base plate and to assess the accuracy of the developed simplified energy-based models. A mesh generation scheme was used for the plate and an iterative scheme was adopted to find the location of the periphery of contact area. This area was found to be more as an ellipse rather than a circle but the corresponding changes in the maximum uplift length and the maximum uplift displacement were found to be negligible. It was shown that uplift is greatly reduced if the outer edge of the plate is restrained against rotation. Although the two approaches are different, they conform in results and conclusions.

In 1990, Yi and Natsiavas [294] presented a finite element model for the seismic response of liquid-filled tanks. They discretized the shell structure using cylindrical finite elements and applied Hamilton's principle in the structural domain to obtain the equations of motion for the coupled fluid-structure system. The above analytical procedure eliminated the fluid hydrodynamic problem by employing the closed form solution for the hydrodynamic response problem, resulting in a compact system of equations of motion. Primary attention was paid to the formulation of the nonlinear base uplift problem. Effects due to the shell and ground flexibility also were included.

Next: Simplified Design Guidelines Up: Evolution of Seismic Analyses Previous: Simplified Models
A. Zeiny
2000-09-06