Advances in Civil Engineering

Volume 2014, Article ID 934284, 17 pages

http://dx.doi.org/10.1155/2014/934284

## Effects of Underground Cavities on the Frequency Spectrum of Seismic Shear Waves

Dipartimento di Ingegneria Civile, Edile e Architettura, Universitá Politecnica delle Marche, Via Brecce Bianche, 60131 Ancona, Italy

Received 22 May 2014; Accepted 18 November 2014; Published 10 December 2014

Academic Editor: Polat Gülkan

Copyright © 2014 G. Lancioni et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A numerical method is proposed to study the scattering of seismic shear waves induced by the presence of underground cavities in homogeneous soils. The method is based on the superposition of two solutions: the solution of the free-wave propagation problem in a uniform half-space, easily determined analytically, and the solution of the wave scattering problem due to the cave presence, evaluated numerically by means of an ad hoc code implemented by using the ANSYS Parametric Design Language. In the two-dimensional setting, this technique is applied to the case of a single cave, placed at a certain depth from the ground level. The frequency spectrum of the seismic shear oscillation on the ground surface is determined for different dimensions and depths of the cave and compared with the spectrum registered without caves. The influence of the cave dimensions and depth on the spectrum amplification is analyzed and discussed.

#### 1. Introduction

In the last decades a growing attention has been paid to the assessment of seismic vulnerability of existing ancient buildings and to the design of safe structures for seismic loadings. To this extent, a large variety of technical codes for designers and engineers have been developed, based on dynamical analysis or equivalent statical calculations. In all of them, crucial data to be taken into account is the frequency spectrum of the seismic oscillation registered at the ground level. The frequency content of a seismic signal depends on many factors such as the topography of the considered zone, the characteristics of the materials which the soil is made of, and the soil stratigraphy. These factors could cause amplifications and increase the seismic hazard for structures built on the nearby ground surface.

Some recent studies [1–8] have pointed out that natural or man-made underground cavities could also be a source of seismic hazards for structures. Earthquakes in a calcareous area carved by underground karstic process, like grottoes, caves, and dolines, can cause the falling in or collapse of the vaults in the former or the removal of debris in the latter. Surface depressions or subsidence of the ground can occur with the resulting danger of collapse for the building above [9]. Some cases of serious damage were found in buildings over man-made cavities after the 1980-earthquake of Atella (Potenza, Italy) [1] and after the same earthquake, the most damaged buildings at Rionero in Vulture were mainly concentrated in areas where there was notable cave density [2]. Gizzi [3] highlight the relationship between the presence of underground cavities and the observed building damage after the 1930-Irpinia earthquake. Similarly, in [4] it was noted that cavities can significantly increase the damage to nearby buildings in the presence of low strength rock. It also seems that the caves can reduce the seismic loading transferred to buildings. Indeed, a small reduction of the peak ground acceleration has been surveyed in areas in which the spatial distribution of the cavities is of some hundred meters [5]. Man-made caves and their relationships with historical buildings were considered in defining their risk level [6]. In the city of Catania seismic response analysis has been performed in proximity of several natural cavities to find that cavities represent a high risk for foundation stability of some buildings [7]. Often ancient buildings are on the surface of soils where many subsurface cavities are present, natural or man-made, excavated over the centuries for housing, defensive purposes, or ritual and religious intents. Italy, for instance, is rich of underground cavities below ancient medieval towns, castles, monasteries, and so forth. These towns are built on essentially man-made underground cavities dating back to many centuries, which are Roman underground water systems of aqueducts and cisterns, medieval tunnels for storage and communications, and hiding places used during the Second World War [8].

To perform a correct seismic structural analyses, a crucial point is the determination of the effective ground surface acceleration due to earthquakes, and, consequently, the finding of the seismic loadings on the buildings. If caves are present underground, then their scattering effects on the seismic wave propagation must be considered for an accurate estimate of seismic accelerations at the ground level.

A first objective of the present study is to propose a numerical technique for determining the seismic frequency spectrum at the ground surface when subsurface caves are present in the soil, for a given frequency spectrum registered at the surface of a soil without caves. To implement this, the ANSYS Parametric Design Language [10] (a fortran-like programming language) was used. An ad hoc code was developed as a computational tool to assist engineers and designers. Indeed, commercial computer software dealing with wave scattering in continuum medium with voids is few or is not comprehensive. On one hand, software for structural analysis may handle sophisticated material nonlinearities and complex geometries, but often it is inadequate for problems of wave diffraction and radiation. On the other hand, codes dealing with seismic wave propagation in soils mainly concentrate on the effects of multilayered stratigraphy. One of these latter codes is the computer program SHAKE [11, 12], which estimates the wave reflection and diffraction occurring at the interfaces between soil layers made of different materials.

Once the computational model was developed, a second objective of the present work is to estimate the influence of the dimensions and depth of an underground cave on the amplitude of seismic wave registered at the ground level, by performing numerical parametrical analysis.

Researches on seismic wave scattering produced by cavities mainly focus on two aspects of engineering interest. A first set of works concentrates on determining the dynamical response of subsurface caves, tunnels in particular, to seismic waves. Different approaches have been followed. In [13, 14], for instance, models based on boundary elements have been developed, while, in [15], both boundary elements and finite elements have been used. In [16], the earthquake-induced strains on tunnels have been evaluated by means of a three-dimensional shell theory, in [17] a multiscale finite element approach has been followed, and, in [18], simplified solutions have been found by modeling tunnels as Timoshenko beams. A second set of studies focuses on the diffracting effects of caves on the wave motion, paying a special attention to the resulting oscillations at the ground level. The present study belongs to this set. A variety of analytical solutions have been found. Only a few papers are mentioned, and the works therein quoted are referred to for more bibliographic references. A review of closed-form solutions for the scattering of pressure and shear waves by a single spherical obstacle has been proposed in [19]. The diffraction problem for shear waves hitting a circular cylindrical cavity placed in an half-space has been solved in [20, 21], and the seismic amplification of surface ground motion above the cavity has been estimated. While in [20] the diffracted solutions of SV-waves have been found by means of Fourier-Bessel series, in [21] the antiplane problem related to the propagation of SH-waves has been solved by using Bessel and Hankel functions. Approximated solutions are determined in [22] by using a hybrid approach which combines the finite element method in the region around the cavity and wave eigenfunction expansions far away from the cavity, in [23] by applying the weighted residual method and in [24] by finite elements. Amplification of the seismic risk for surface structures on soils with voids has been investigated in [25]. In it, the response of a rigid embedded foundation to antiplane SH-waves diffracted by a cylindrical cavity has been evaluated by using Bessel functions. In [26], the influence of SV-waves in the surface motion has been investigated numerically by means of the explicit finite difference program FLAC [27]. The diffraction of waves by cavities has been studied in the real case of the town of Castelnuovo (Italy), characterized by many underground cavities, which was stricken by the violent L’Aquila earthquake in 2009.

Here we consider only SV-waves, with plane wavefront and upward direction of propagation, which are the most demanding for ancient buildings since they transmit inertial horizontal forces. The model is restricted to the two-dimensional setting. A homogeneous linearly elastic half-plane with a single void placed at a certain depth from the ground is considered. Since the problem is assumed linear, the strategy we follow to solve it is based on the superposition of two solutions.(i)The first one is the so-called* free-field problem*, that is, the problem of wave propagation in a homogenous half-plane. For this, the solution is available analytically.(ii)The second one, named* diffracted problem*, accounts for the perturbation brought by the presence of caves to the wave propagation. This problem consists in determining the motion due to a certain distribution of forces applied on the cave boundaries, determined as function of the free-field solution. Since this problem cannot be solved analytically, it is approached numerically, by means of finite elements. A semicircular computational domain is considered, and absorbing conditions are assigned at the artificial boundary to avoid wave reflection. The two main techniques for absorbing outgoing waves are the high-order absorbing boundary conditions and the perfectly matched layer (see [28, 29] for a comparison of these methods and the references therein quoted for detailed descriptions of them). Here, the first-order absorbing boundary conditions are considered [30], avoiding more complex higher-order conditions. First-order conditions perfectly filter waves which hit the boundary with null incident angle. Since in the diffracted problem waves generated at the cave boundary propagate with circular wave fronts, hitting the artificial boundary with very small incident angles, then first-order conditions guarantee satisfactory accuracy, providing that a sufficiently large radius of the semicircular domain is considered. Once the diffracted problem is numerically solved, the solution of the global problem is obtained as sum of the diffracted and free-field solutions.

In this paper, the above described strategy is applied to the problem in the frequency domain. The frequency spectrum of seismic shear oscillations is determined at the ground level, as required for engineering applications where the frequency content of shear oscillations is a fundamental input data for seismic analyses of existing or new buildings. Several simulations are performed by considering different dimensions and depth of an arch-shaped cave. For each simulation, the seismic oscillation registered during the Umbria-Marche earthquake of 1997 is assumed as the solution of the free-field problem. The amplitude of its frequency spectrum is evaluated, and then the corresponding diffracted problem is solved by using finite elements. Finally, the global solution is estimated, and the amplitudes estimated at the ground level, right above the cavity, are compared with those of the free-field case. Depending on both dimensions and depth of the cave, frequency ranges are found in which oscillation amplitudes are amplified and frequency intervals are found in which they are reduced. A detailed description of the found spectra is made, and the differences with the free-field spectrum are discussed.

The paper is organized as follows. In Section 2, the problem is formulated. The material characteristics of soil and the cave geometry are defined, and the free-field solution is determined. In Section 3, the strategy used to solve the problem is described first in the time domain and then in the frequency domain. Section 4 deals with the absorbing boundary conditions assigned at the artificial boundaries of the computational domain. The radius of the semicircular computational domain is determined by means of numerical analyses. In Section 5 the numerical results are presented and discussed. Concluding remarks are drawn in Section 6.

#### 2. Problem Statement

The scattering problem of seismic waves caused by underground cavities in soils is formulated in the two-dimensional setting. The soil is represented by a half-plane, homogeneous and linearly elastic, with mass density kg/m^{2}, Young modulus N/m^{2}, and Poisson ratio . These values characterize a real soil where caves are present [31]. In this medium, dilatational and shear waves propagate with velocities m/s and m/s, respectively. They are related to and by
It is assumed that a cave is placed in the soil at a certain depth from the ground level and has the shape drawn in Figure 1. Then, it is suppose that seismic shear waves propagate in the soil from the bottom to the top, with wave velocity . Figure 1 sketches a geometrical scheme of the problem. To represent points and vector by components, the orthogonal Cartesian frame , depicted in Figure 1, is used.