Dynamic Stress and Displacement Fields on Concrete Gravity Dams due to Harmonic Ground Motion

Abstract

The partial differential equations (PDE) of equilibrium governing the natural vibrations of concrete gravity dams were derived in this work such that the fluid structure interactions were accounted for. The displacement formulation is a system of two coupled PDE in two unknown displacement components. For seismic ground motion assumed to be horizontal harmonic motion whose amplitude and period are known, the system of two coupled PDEs were solved subject to the boundary conditions using the method of undetermined parameters. In applying the method of undetermined parameters to the PDE, displacement shape functions constructed to satisfy the displacement boundary conditions were used in assuming the trial dynamic displacement fields in terms of two unknown parameters that were determined by substitution into the governing equations. Conditions for the trial dynamic displacement fields to be solutions to the governing PDE were sought by solving the resulting system of equations. The problem reduced to an algebraic eigenvalue eigenvector problem which was solved for nontrivial cases to obtain the characteristic frequency equation and the eigenvalues. Modal superposition technique was employed to obtain the general solution for the displacement fields. The use of the displacement boundary conditions at the upstream face and Fourier series theory yielded the dynamic displacement field components, and the dynamic stress fields. The maximum value of the hydrodynamic pressure distribution on the upstream face is found to occur at the bottom of the dam and is found mathematically to be a convergent series of infinite terms. The maximum hydrodynamic force was calculated by integration of the hydrodynamic pressure distribution over the upstream face of the dam, and found to be a convergent series. Values of the maximum hydrodynamic force computed in this work agree with solutions from the technical literature.