Evolution of digital computers and associated numerical techniques have significantly enhanced solution capabilities of complex problems. The first use of a digital computer in analyzing anchored liquid storage tanks was completed by Edwards [51]. He employed the finite element method and a refined shell theory to predict seismic stresses and displacements in a vertical cylindrical tank having a height to diameter ratio smaller than one. This investigation treated the coupled interaction between the elastic wall of the tank and the contained liquid. The tank cross section was assumed to be restrained against cross section distortions. Shaaban and Nash [248] undertook a similar research concerned with the earthquake response of cylindrical elastic tanks using the finite element method. Shortly after, Balendra and Nash [14] offered a generalization of the analysis by including an elastic dome on the tank. Fenves [61] used a mixed displacement-fluid pressure formulation for the fluid, and a standard displacement finite element formulation for the structure.

A different approach to the analysis of flexible containers was developed by Veletsos [284]. He presented a simple procedure for evaluating hydrodynamic forces induced in flexible liquid filled tanks. The tank was assumed to behave as a single degree of freedom system, to vibrate in a prescribed mode and to remain circular during vibrations. Hydrodynamic pressure distribution, base shears and overturning moments corresponding to several assumed modes of vibration were presented. Later, Veletsos and Yang ([282], [283]) estimated maximum base overturning moment induced by a horizontal earthquake motion by modifying Housner's model to consider the first cantilever mode of the tank. They presented simplified formulas to obtain the fundamental natural frequencies of liquid filled shells by the Rayleigh-Ritz energy method. Another approach was attempted by Natsiavas [200]. He expanded the structural displacements in appropriate series forms which involve both rigid body and flexible components. The latter components were expressed as linear combinations of terms, each of which is a product of a function with assumed spatial dependence and an unknown time-dependent function. These time functions were then determined from solving the equations of the fluid-structure system, which were set up by employing Hamilton's principle.

In 1980 and 81, Haroun and Housner [106] used a boundary integral theory to drive the fluid added mass matrix, rather than using the displacement based fluid finite elements. The former approach substantially reduced the number of unknowns in the problem. They conducted a comprehensive study ([97], [98], [100], [101], [102], [103], [104], [106]) which led to the development of a reliable method for analyzing the dynamic behavior of deformable cylindrical tanks. A mechanical model [104], which takes into account the deformability of the tank wall, was derived and parameters of the model were displayed in charts to facilitate the computational work. The model may be applied to predict the maximum seismic response by means of a response spectrum. It received wide spread application because the previous models were either too complicated to be used in the design or too simple to give accurate results. Later, Haroun included more complicating effects in his analyses, such as the effect of out of roundness on the dynamic response of flexible tanks, the effect of initial hoop stress on the -type modes and the soil-structure-fluid interaction ([83], [87], [88], [92], [105], [188], [189]).

The boundary integral technique was also employed by Williams and Moubayed ([289], [290]) to investigate the response of liquid storage tanks. They utilized a fluid Green's function to reduce the fluid domain to a line integral of the velocity potential on the surface of the structure. They calculated the hydrodynamic pressure distribution on a rigid submerged cylindrical tank subjected to a horizontal or vertical harmonic ground excitation. In a following work [288], they expressed both the structural and fluid axisymmetric motions in terms of appropriate Green's functions that lead to a pair of coupled line integral equations for the fluid velocity potential and its normal derivative on the walls of the tank. They studied the influence of the frequency of the ground excitation and the various geometric and material parameters on the hydrodynamic pressure distribution and the associated dynamic response of liquid storage tanks when subjected to high frequency vertical ground motions.

The finite element method combined with the boundary element method was used by several investigators, such as Grilli [72], Huang [114] and Kondo [142], to investigate the problem. Hwang ([119], [120]) employed the boundary element method to determine the hydrodynamic pressures associated with small amplitude excitations and negligible surface wave effects in the liquid domain. He obtained frequency-dependent terms related with the natural modes of vibration of the elastic tank and incorporated them into a finite element formulation of an elastic tank in frequency domain. Generalized displacements were computed by synthesizing the complex frequency response using the Fast Fourier Transform procedure.

The vibrations of a thin-walled cylindrical shell filled with an ideal liquid were investigated by Goncalves [70]. He used modal expansions for the displacements of the shell and expanded the velocity potential function in terms of harmonic functions which satisfy the Laplace equation. The Galerkin method was used to reduce the problem to a system of coupled algebraic nonlinear equations for the modal amplitudes. All of these studies showed that seismic effects in a flexible tank may be substantially greater than those in a similarly excited rigid tank.