Abstract
We revisit previous results in small-strain One-dimensional Site Response Analysis of heterogeneous soil deposits. Specifically, we focus on sites whose shear modulus distribution is described by means of a power law that yields zero stiffness at the free surface. First, we show that in some cases (which we characterize in detail) considerations of energy finitude should prevail over considerations of vanishing tractions at the free-surface, as these may pose acuter constrains. We re-evaluate previous contributions in light of this result. Second, we analyze the previously-reported occurrence of “energy accumulation in upper layers”, providing a physical explanation for it. In passing, we supply estimates of the natural frequencies, and compare these with our previous results.
Highlights
One-dimensional site response analysis is a popular engineering approach to estimation of the ground deformation caused by seismic events
Impinging S waves that can be modeled as a forced-displacement boundary condition at the bedrock level
It is a known fact that the stiffness displayed by the soil is mediated by the overburden pressure [3], in general, the deeper soil layers are more difficult to deform than the upper ones, a fact that is translated into the models as a increasing stiffness with depths
Summary
One-dimensional site response analysis is a popular engineering approach to estimation of the ground deformation caused by seismic events. An alternative justification to working in frequency domain is provided by invoking Random Vibration Theory as the framework for the analysis [2] Such a configuration combines relative simplicity, what make it suitable to be tackled with analytical techniques, with relevance for practical engineering purposes. The stiffness of the first infinitesimal slice of soil at the ground would present no stiffness if one assumes that the soil behaves as a granular flow in absence of the confining pressure (soil column weight) This feature (“stiffnessless” of ground surface) leads to some striking consequences. Where z is the coordinate stretching from the free surface to the rigid base, ρ represents the density of the soil (which is assumed to be constant), the seismic S-wave excitation is modeled as a uniform displacement at the soil base Xgei t, and μ(z) represents the variation of shear modulus with depth. For the sake of clarity, let us restate that the time variation comes given by ei t, both Xg and ut are the amplitudes that accompany the phase: Xg is a positive real number, whereas ut represents a complex number whose modulus is the total displacement magnitude and whose argument represents the time shift between load (input) and displacement (response, output)
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