Variational Solutions for Externally Induced Sloshing in Horizontal-Cylindrical and Spherical Vessels

A mathematical model is developed for calculating linear sloshing effects in the dynamic response of horizontal-cylindrical and spherical liquid containers under external excitation, with emphasis on earthquake excitation. The velocity potential is expressed in a series form, where each term is the product of a time function and the associated spatial function. Because of the configuration of the containers, the associated spatial functions are nonorthogonal and the problem is not separable, resulting in a system of coupled nonhomogeneous ordinary linear differential equations of motion. The solution can be obtained through either direct integration or modal analysis. Particular emphasis is given on the rate of convergence of the solution. The cases of half-full cylinders and spheres are examined in detail, where explicit expressions for the coefficients of the governing equations are derived. Using the proposed methodology, sloshing frequencies and masses are calculated rigorously for arbitrary liquid height of horizontal-cylindrical or spherical containers, and the response under two characteristic seismic events is obtained. The results describe the linear dynamic response of such containers and can be used for an efficient seismic analysis and design of industrial pressure vessels.