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## Simplified Models

Based on simplified theoretical investigations, many researchers have attempted to develop simplified models for unanchored tanks. Most of these investigations have focused attention on the behavior of the bottom plate which is a governing factor for the behavior of unanchored tanks. It started in 1977 when Clough [36] proposed a simplified model for uplifting of unanchored tanks, but he ignored the load carrying capacity of the tank bottom plate. It was assumed that the bottom plate remains in contact with foundation on a circular area of a radius slightly less than the radius of the tank, and that the edge of shell rests on an arc of the tank's perimeter of an unknown central angle. A geometrical relation was found between the central angle and the radii of the contact area before and after uplift. Total overturning moment which causes the plate to uplift consists of two components: the moment exerted on the tank wall by the liquid and the additional moment on the tank bottom. The values of these moments were found from Housner's analysis of tanks with rigid walls [272]. The two unknowns of the problem (the maximum compressive stress in the tank shell and the central angle of the contact area) were found by solving two nonlinear algebraic equations which govern both global vertical force equilibrium and global moment equilibrium. One disadvantage of this model is that it does not take into account the flexibility of either the tank wall or the bottom plate. Furthermore, it neglects the variation of dynamic pressure on the bottom plate and uses a constant value equal to the static pressure. In 1978, Wozniak and Mitchell [292] suggested a more realistic model for uplifting by including the flexural stiffness of the bottom plate, and this analysis was introduced in the AWWA D100 [6] and the API 650 Standards [7]. It was assumed that the contact area of the bottom plate with the foundation is a segment of an unknown central angle. Flexibility of the tank wall was not considered but the elastic behavior of the bottom plate was taken into account. The base plate was represented by a strip of a unit width in the circumferential direction because the relevant uplift region is assumed to be an annular ring of a width much smaller than the radius of the tank. The strip acts as a beam resting on a rigid foundation subjected to a liquid pressure and lifted up by a vertical force at its free end. The maximum value of the force that can be carried by the beam is calculated by invoking two plastic hinges: one at its free end and one at an intermediate point in the uplifted portion of the beam. Because the wall thickness is usually larger than the thickness of the bottom plate, the assumption of a plastic hinge at the edge of the bottom plate is justified. Assuming no restraining effects from the shell and no membrane stress in bottom the plate, relations between the thickness of the plate, the plastic stress, applied distributed loads and the uplifted length can be found explicitly. If the tank experiences uplift, two forces resist such deformation, namely, the weight of the roof and the shell, and the weight of liquid that will be lifted up. By writing the equations of equilibrium of vertical forces and moments, the maximum compressive stress and the central angle can be found as in previous model. In 1986, Leon and Kausel [151] proposed some modifications to Wozniak and Mitchell's model. They concluded that this model, which forms the basis of provisions of API 650 Standard, can lead to a significant underestimation of maximum compressive stresses in the shell under the condition of moderate shell uplift, and overestimation of the contribution of fluid weight in resisting lift-off.

Moore and Wong [184] modified parameters defining the maximum width of the uplifted strip of the tank base and the associated compressive stresses in tank walls from those specified in the API 650 Standard which gives conservative values. They collected an extensive set of damage data from the 1964 Alaska, the 1971 San Fernando, the 1978 Miyagi-Ken-Oki and the 1980 Livermore earthquakes as well as past experimental results and reported that the correlation between predicted seismic resistance using the modified API code model and the observed damage was good, and that unanchored tanks with an aspect ratio of H/D between 0.6 and 1.4 were more prone to damage than were broader or more slender tanks with the same depth of contents. In the same year, 1984, Sakai, Ogawa and Isoe [240] discussed the elastic behavior of a cylindrical liquid storage tank under horizontal, vertical and rocking motions. The shell was assumed ideally cylindrical, and nonlinear effects of the bottom plate uplift were neglected. They conducted experiments on model tanks to verify their analysis, and presented simplified procedures for analyzing fluid response to each type of motion. In 1986, Manos [172] proposed an empirical design approach regarding the tank-wall earthquake stability for unanchored cylindrical liquid storage tanks. Based on experimental evidences, he assumed an empirical compressive axial membrane stress distribution with a maximum value equal to 75% of theoretical buckling stress of a uniformly compressed perfect cylinder. By equating the tank resisting overturning moment calculated from assumed stress distribution with the earthquake induced overturning moment obtained from modified Housner's model considering impulsive response only, he found a limit impulsive acceleration coefficient that uplifting tanks could withstand. However, the complex dynamic uplift behavior of the bottom plate and its stiffness were ignored. In addition, this approach was only a design approximation and was not based on any rigorous analytical model. Manos concluded that existing approaches were unreliable in predicting damages, and that they underestimated the actual limit impulsive acceleration that tanks might withstand, and recommended that a realistic analytical treatment of nonlinear dynamic response mechanism of uplifting tanks should be developed.

In 1988, Natsiavas and Babcock [199] presented an analytical model for the response of an unanchored tank subjected to horizontal base excitation by using Hamilton's principle. Base plate uplift was modeled as a rotational nonlinear spring whose characteristics were obtained from previous static tilt tests. It was assumed that the tank rests on a flexible ground but with a rigid foundation, and equivalent springs were used for formulation of ground flexibility. They concluded that it can be significantly unconservative to calculate loads for an unanchored tank by assuming the tank to be anchored. In another work, Natsiavas [198] showed that base uplift causes a dramatic reduction in the deflective beam-type stiffness of a tank which in turn reduces the tank response frequency and changes developed hydrodynamic loads significantly. He also provided an explanation for some great qualitative and quantitative differences in the behavior of a tank, resulting from its base fixity condition alone. He also showed comparisons between numerical results from his analysis with experimental data. In a following work [197], he presented another analytical model for the problem. He solved the hydrodynamic problem in closed form for the most general motion of the structure. Then, he applied Hamilton's principle to derive the equations of motion of the system. The uplifting behavior was modeled by an appropriate rotational spring placed between the foundation and the bottom of the tank. Effects due to ground flexibility, shell flexibility and liquid sloshing were also included. Using this model, results were obtained and compared with experimental data. Buckling of scale model tanks during experiments was also investigated. Furthermore, in 1989, he presented a couple of simplified models [196] determining the hydrodynamic loads during the seismic response of tall unanchored liquid storage tanks. In both models, the general procedure starts by solving the fluid response problem in closed form to eliminate all the unknowns associated with the hydrodynamic problem but the sloshing ones. In the first model, the geometrical discretization was performed by applying Hamilton's principle. In the second model, the system, parameters were identified by requiring the same base loads when the real system and the model were subjected to the same base motion.

In 1988, Peek [220] simplified a method of analysis for static lateral loads based on the assumption that the restraining action of the base plate can be modeled with equivalent nonlinear springs. The displacements was decomposed using Fourier analysis. The solutions were compared with those from experiments and current U.S. design analysis methods. In a following work, Peek et al [217] described a simplified approach for determining the extent of the elephant foot buckling for ground supported unanchored liquid storage tanks subjected to seismic overturning moments. He reported that the ultimate seismic overturning moment that can be withstood by a tank is higher than the overturning moment at which the elephant foot buckling first occurs. In 1989, Fischer [64] considered analytically the frictional contact behavior of extensible and flexible strips on rigid grounds and loaded by a transversal pressure. He also investigated the uplift behavior of the strip. In 1994, Malhotra and Veletsos ([165], [166]) simplified the unanchored tank-liquid system to one degree of freedom system with rotational spring representing the rocking resistance of the base plate. The relationship between the base moment and the spring rotation was established by considering uniformly loaded, semi-infinite, prismatic beams that are connected at their ends to the cylindrical tank wall. However, the flexibility of the tank itself was ignored.

Next: Numerical Analyses Up: Evolution of Seismic Analyses Previous: Experimental Investigations
A. Zeiny
2000-09-06