The study of the seismic response of inhomogeneous soil deposits, underlying rigid bedrock, has traditionally been framed as a one-dimensional wave propagation problem in a linear viscoelastic medium. When it comes to modelling soil inhomogeneity, the current trend to model the continuous variation of soil stiffness with depth employs a ‘generalised parabolic function’. Exact solutions have already been obtained, but these come in terms of Bessel functions, which muddle the interpretation of results and obstruct assessment of the parameters’ influence. This also hinders the definition of an ‘equivalent homogeneous’ soil deposit (a relevant concept utilised in many seismic codes). It is shown herein that the so-called inhomogeneity factor plays a secondary role in determining the response in many cases; thus straightforward guidelines are suggested for simplifying the problem, leading to elementary scaling relations. The scalings provide simple yet meaningful relations that reveal and explain the fundamental traits of the dynamic behaviour of these systems.
The technique referred as Geometrical Optics entails considering the wave propagation in a heterogeneous medium as if it happened with infinitely small wavelength. This classic simplification allows to obtain useful approximate analytical results in cases where complete description of the waveform behavior is virtually unattainable, hence its wide use in Physics.This approximation is also commonly termed Ray Theory, and it has already been thoroughly applied in Seismology. This text presents an application of Geometrical Optics to 1D Site Response (1DSR): it is used herein to, first, explain and elucidate the generality of some previous observations and results; second, to partially settle an open question in 1DSR, namely "what are the equivalent homogeneous properties that yield the same response, in terms of natural frequencies and resonance amplitude, for a certain inhomogeneous site?", provided few assumptions.
The technique referred as Ray Approximation treats wave propagation in a heterogeneous medium at the infinitely small wavelength limit. This classic simplification allows to obtain useful approximate analytical results in cases where complete description of the waveform behavior is virtually unattainable, hence its wide use in Physics. In Seismology, this approximation has been widely applied. This text presents an application in one-dimensional Site Response (1DSR) analysis: it is used herein to, first, explain and elucidate the generality of some previous observations as to the use of the harmonic mean of a shear-wave velocity profile to represent the global behavior of a site; and second, to partially settle an open question in 1DSR, namely “what are the equivalent homogeneous properties that yield the same response, in terms of natural frequencies and resonance amplitude, for a given inhomogeneous site?”, provided few assumptions, chiefly, considering excitations of high enough frequency.
The problem of estimating seismic ground deformation is central to state-ofpractice procedures of designing and maintaining infrastructure in earthquakeprone areas. Particularly, the problem of estimating the displacement field in a soft shallow layer overlying rigid bedrock induced by simple SH wave excitation has been favored by engineers due to its simplicity combined with inherent relevance for practical scenarios. We here derive analytical, accurate estimates for both the fundamental frequency and the amplitude of the first resonant mode of such systems by applying an intuitive argument based on resonance of single-degree-of-freedom systems. Our estimates do not presuppose a continuous velocity distribution, and can be used for fast assessment of site response in seismic hazard assessment and engineering design. On the basis of the said estimates of fundamental frequency and amplitude, we next propose a novel definition of "equivalent homogeneous shear modulus" of the inhomogeneous deposit; and we show that the response of the fundamental mode of these systems is governed by the mechanical properties of the layers closer to the bedrock. We finally discuss the validity of our argument, and evaluate the accuracy of our results by comparison with analytical and numerical solutions.
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