Seismic Response of Liquid Storage Tanks

Abstract

Liquid storage tanks are used worldwide to contain liquids such as drinking water, fuel and chemicals. Loss of containment during a seismic event can add to an earthquake’s destructiveness by posing a fire hazard, spilling toxic substances or reducing the supply of drinking water. It is therefore important that they remain operational at a pre-defined level of functionality during and after an earthquake event. Two effects that are recognized to play a role in the seismic behavior of liquid storage tanks are fluid-structure interaction and soil-structure interaction. Numerical models have the ability to describe the dynamic behavior of each domain accurately, but they tend to be time consuming compared to many of the available simplified analytical models. While the latter can be useful for design purposes, they do not necessarily yield accurate solutions to the complex system at hand. In this thesis, a semi-analytical model is put forward based on previous work by Canny (2018) and Molenkamp (2018) that combines both fluid-structure and soil-structure interaction with a dynamic substructuring approach. The model offers a computational efficiency and ease of use comparable to the aforementioned analytical models, but the only sacrifices to accuracy are the limitations imposed by the underlying assumptions and a number of discretizations. The tank, soil and fluid domains are treated separately. The tank is considered with Love's thin shell theory and the soil as a visco-elastic continuum. The fluid is assumed to be incompressible, irrotational and inviscid, allowing the application of linear potential theory. The solutions to the homogeneous equations of Love's shell theory, describing the behavior of the tank's bottom plate and wall segments, are found as eigenfunction expansions. By satisfying the boundary and interface conditions of the different tank parts, an eigenvalue problem is formulated that leads to the eigenfrequencies and mode shapes of the tank. The fluid velocity potential is rewritten as the superposition of three fluid velocity potentials that each satisfy a set of conditions regarding the velocity continuity at the wall and at the plate, and the free surface condition at the fluid surface. The velocity potential in addition needs to satisfy Laplace's equation and solutions are found again as an expansion of eigenfunctions that represent the fluid's modes. For the soil, Green's influence functions for uniform horizontal and vertical loads on a circular area at the soil surface have been used to assemble a dynamic soil stiffness matrix. The influence functions are again in the form of an eigenfunction expansion that is the solution to the homogeneous wave equation describing the soil's motions. Through satisfaction of a set of conditions at the fluid-structure and soil-structure interfaces, respectively velocity and stress continuity, and displacement and stress continuity, a final set of equations is obtained. The only unknowns remaining in this set are the modal amplitudes that after solving can be used to linearly combine the mode shapes to give the full dynamic behavior of the tank-fluid system. Motions and stresses at the soil surface can be found with the dynamic soil stiffness matrix. The dynamic soil-structure interaction is taken into account in the model by considering all wave fields present in the soil, namely the free-field (or incident), the scattered and the radiated wave field. Subsequent satisfaction of the soil-structure interface conditions ensures that soil-structure interaction is properly considered, and it yields a simple result in which the usually known free-field wave field can directly be applied in the excitation term of the plate's equations of motion.
Results are obtained in the frequency domain and can be transformed to the time domain with the inverse Fourier transform. Output can be obtained similar to the output of FE models with stresses and displacements at each location of the tank domain. In the fluid domain, velocities can be found as well as related variables, such as the fluid pressures at the fluid-tank interfaces and the fluid elevation at the fluid surface (sloshing). Regarding the soil, stresses can be obtained at the soil-structure interface and displacements at the soil surface. Besides, the model can give more insight in the effects of SSI, compared to a tank-fluid system founded on a rigid soil. Limitations of the model include the inability to model nonlinear effects, as a result of the linearity of the model, so that nonlinear failure mechanisms as buckling cannot be modeled, or nonlinear stress-strain relations. At the same time, nonlinear effects are not extensively covered in the other available analytical methods either. With regard to engineering practice, roofs, ring stiffeners and anchors are commonplace, but have yet to be included in the model. Similarly, the capacity to model multiple soil layers would bring the model a step closer to reality. Besides more accurate modeling of the soil composition at a greater depth, it would admit a more realistic model of the soil directly underneath the tank, which is often improved in case of initially unsuitable grounds. Before the model can be applied in practice however, it needs to be validated, for example with a FE model or experimentally. To increase the model's competitive advantage of computational efficiency, measures can be taken to improve it. The greatest improvements can be made in the assembly of the dynamic soil stiffness matrix by reducing the number of elements of the soil-structure interface. With the applied axisymmetric discretization, many of the matrix entries are the same, so that only a small number actually needs to be computed. However, a large number of very small elements is concentrated in the center of the soil-structure interface that does not add to the accuracy of the solution. Other discretizations could reduce computation time by reducing the number of elements and thereby the size of the dynamic soil stiffness matrix.