The dynamic response of liquids has significant influence on the response of their containers. Inappropriate approximation of the liquid motion may lead to major errors in estimating the seismic response of the containers. The liquid pressures and the impact forces form the measurable level of the energy transferred to the tank shell. In addition, the motion of the tank wall is the primary source for the liquid energy. Since this energy transfer occurs simultaneously throughout the liquid boundary, it is essential in the finite element analysis of such problems to use a model that effectively deals with the coupling between the liquid and the tank wall.

The equations of motion of a liquid may be formulated by two different approaches, corresponding to the two ways in which the problem of determining the motion of a liquid mass, acted on by given forces and subjected to given boundary conditions, may be viewed. The Eulerian formulation is obtained by considering the object of our investigations to be the knowledge of the velocity, pressure and density at all points of space occupied by the liquid for all instances. On the other hand, the Lagrangian form is obtained by considering the object to be the determination of the history of each particle. Detailed discussions of the two forms may be found in [146]. In the current investigation, a Lagrangian description of the structure's motion is utilized, which makes it necessary to use a Lagrangian description of the liquid-structure interface in order to enforce compatibility between the structure and liquid elements. The continuity equation in the Eulerian form is utilized inside the liquid domain to mathematically describe the liquid motion inside the tank.

The liquid in this analysis is considered to be inviscid, irrotational and incompressible. Such simplifying assumptions allow displacements, pressures or velocity potentials to be the variables in the liquid domain. The displacement-based liquid elements may be easy to incorporate in finite element programs for structural analysis and simplify the enforcement of the liquid-structure interface constraints. However, such elements require two or three degrees of freedom per node. In addition, this approach is not well suited for problems with large liquid displacements and requires special care to prevent zero-energy rotational modes from arising. Alternatively, using pressures or velocity potentials as the unknown degrees of freedom requires only one degree of freedom per node inside the liquid domain, which significantly reduces the computational cost of the analysis, and adequately represent the physical behavior of the liquid. The latter approach is used in this investigation.

- Variational Principles of the Liquid-Structure Interaction Problem
- Finite Element Discretization
- Computer Implementation and Testing
- Two-Dimensional Examples
- Example 1: Linear Response to Sinusoidal Ground Excitation
- Example 2: Nonlinear Response to Sinusoidal Ground Excitation
- Example 3: Effect of Tank Width on the Response to Sinusoidal Ground Excitations
- Example 4: Nonlinear Response to Mexico City Earthquake
- Example 5: Nonlinear Response to El Centro Earthquake
- Example 6: Free Surface Wave Breaking Due to Resonance
- Example 7: Free Surface Wave Breaking Due to High Level of Excitation

- Three-Dimensional Examples

- Two-Dimensional Examples