# Seismic Analysis of Gravity Dams and Tanks

Institution:
IUSS Pavia
Specialization:
EE - Earthquake Engineering
Term:
Fall 2006
Teacher(s):
CARLOS PRATO
Credits:
6
Date (from - to):
20/11/2006 – 15/12/2006
Theory covered: - Linear formulation of the compatibility and equilibrium equations for ideal compressible fluid. Boundary conditions for the free surface of the fluid. Linear approximation for the gravity surface waves. - Formulation in terms of the hydrodynamic pressure (compatibility method). - Formulation in terms of the displacements (stiffness method). - Finite Element solutions based on nodal values of the hydrodynamic pressure (acoustic elements) - Finite Element solutions based on nodal displacements (stiffness elements). - Natural vibration modes of cylindrical and rectangular prismatic tanks associated with fluid compressibility and with gravity surface waves (sloshing modes). - Closed form solution of the wave equation for a compressible fluid stratum subject to imposed horizontal acceleration at a vertical boundary, and to a vertical acceleration at a horizontal boundary. - Simplifications for the incompressible fluid. Added mass concept and calculation. - General formulation of an elastic gravity dam interacting with an infinite reservoir. - Solution of the fluid-structure interaction problem based on a two-stage procedure. Calculation of the interaction degrees of freedom on a reduced subspace and calculation of the stresses in the dam and foundations in the second stage. - Analysis of the effect of water compressibility on the seismic response of a gravity dam as function of the dam height. Applications: Applications of the theory will be carried out using a Matlab Tool Kit, and FE approximations of 2-D problems for rectangular and triangular elements. The following sample problems will be presented and proposed: - Calculation of the natural vibration modes and frequencies for rigid circular cylindrical and rectangular tanks. Comparison of closed the form solution with F.E. approximations based on simple acoustic elements. - Estimation of sloshing mode amplitude due to harmonic excitation of tank walls.. - Calculation of the added mass of the fluid contained by a cylindrical and rectangular tank structure. Effect of the boundary condition at the top surface of the fluid depending on the assumption concerning the value of the pressure or of the acceleration of that top surface. Effect of the tank height/width aspect ratio on the added mass of the fluid on the mass of the structure. - Hydrodynamic pressure transfer function of a vertical-face concrete gravity dam. Thresh-hold frequency for energy radiation from the dam face through the fluid.