A refined analysis of sloshing effects in seismically excited tanks

Sloshing in terms of liquid surface displacement in vertical liquid-filled cylindrical tanks under earthquake excitation is a well studied phenomenon. Various design rules exist for liquid storage tanks to sustain the corresponding liquid pressure due to seismic excitation and to take into account the necessary freeboard. However, usually the sloshing motion is considered under the assumption of a rigid tank with an earthquake excitation at the base circle. The arguments used so far in justifying this assumption are of rather qualitative but not of quantitative nature. Since it is important to have a quantitative measure of that which is neglected, it is the intention of this paper to show that this engineering approach is based on rigorous theoretical quantitative results. Therefore, in this paper coupling of sloshing with the deformations of a flexible tank wall during earthquake excitation is investigated in a refined analysis. In contrast to former papers which have studied the negligible influence of the wall deformations due to sloshing itself, in this paper the more important coupling including the wall deformations caused by the impulsive effect of the contained liquid is taken into account. An analytical procedure is presented which allows one to study explicitly the influence of the wall deformations on both the liquid pressure and the surface elevation for typical wall deformation shapes, i.e. vibration modes. From the rather complex mathematical derivations a simple formula is drawn which enables the engineer to get a quick guess of the magnitude of the influence of the wall deformations on the convective pressure contributions due to sloshing and hence to decide whether or not the assumption of a rigid tank wall is suitable. It is shown that for tanks made of less stiff materials, such as for instant polymers, this rigid wall assumption which is suitable for steel tanks may become questionable.