The evaluation of the structural response of concrete dams to earthquake is carried out by means of numerical simulations generally with the finite element method (FEM), which allow to reproduce with good accuracy the real dynamic behaviour of the system consisting of the dam, the reservoir and the foundation. Different methods of analysis can be adopted, characterized by increasing complexity: the simplest methods, based on some simplifying physical assumptions, generally provide a faster, conservative and more comprehensible response; on the other hand, the most complex methods, although computationally expensive, allow to successfully deal with physical situations that often the simplest methods are not able to represent, providing a more accurate and realistic description of the structural behaviour and reducing the excessive conservativity of the results.

An accurate method for the seismic analysis of dam-foundation-reservoir systems should be able to consider the three-dimensionality of the problem, the semi-unbounded size of the foundation, the non-linear behaviour of the system and the dynamic interactions of the dam with the foundation and the reservoir. An advanced approach, to be adopted in the frame of time-history linear or non-linear analyses and able to face all the aforementioned complexities, was recently implemented and tested by RSE: the approach, indicated by the acronym SAM-4D (Seismic Advanced Model for Dams), allows to appropriately simulate the propagation of seismic waves in a realistic massed foundation, considering its semi-unbounded extent.

Artificial non-reflecting (or absorbing) boundaries are used to delimit the semi-unbounded foundation and effective earthquake forces (computed with reference to the elastic wave vertical propagation theory) provide the seismic motion. The Pine Flat Dam, used in the frame of the ICOLD 15th International Benchmark Workshop on Numerical Analysis of Dams, is considered as a case study to perform analyses of increasing complexity, starting from the pseudo-static analysis up to the non-linear time-history analysis using the SAM-4D model.