Geometric analysis of trapezoidal hills subject to vertically incident SH waves
Abstract
Topographic effects have been shown to play a significant role on the local ground response during earthquakes. However, due to the large number of involved parameters the problem is rarely considered in seismic design regulations. Recently, there has been a tremendous development by the engineering community, regarding methods and computational infrastructure to address the problem via numerical simulations.
Although numerically based models may give accurate results when fed with appropriate field data, the obtained solutions are still very limited and strongly dependent on unknown factors like the input excitation. Therefore, there is a clear need to develop strong conceptual understanding allowing practising engineers to arrive at first order approximations, useful to validate complex numerical solutions. In this work we explore the use of purely geometrical methods in the determination of the dynamic response of trapezoidal geometries to vertically incident horizontally polarized shear waves.
The geometries may be considered representative of hills or earth embankments, depending on its characteristic dimensions. The hill response is first found with a frequency domain based boundary element code and the results are later analysed using a geometric approach, where the solution is partitioned into incident and reflected rays, forming the incoming or optical field, and the diffraction contribution.
This last term is obtained with a technique available from the literature. The analysis corresponding to the optical field, reveals that there are only 5 possible scenarios or different solutions and that any given hill can be classified into one of these five possible cases. Depending upon the dimensionless frequency of the problem (relating incident wavelength to the hill characteristic dimension), the solution is found to be governed by the optical solution or by the contribution from the diffraction terms. The results are first presented in terms of frequency amplitude functions since that description facilitates the analysis by geometric methods, however for completeness, the resulting transfer functions are later used to obtain results in the time domain representative of typical numerical solutions as the ones derived with commercial computational software.
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