^{1}

^{2}

^{1}

^{2}

The basic idea of a seismic barrier is to protect an area occupied by a building or a group of buildings from seismic waves. Depending on nature of seismic waves that are most probable in a specific region, different kinds of seismic barriers can be suggested. Herein, we consider a kind of a seismic barrier that represents a relatively thin surface layer that prevents surface seismic waves from propagating. The ideas for these barriers are based on one Chadwick's result concerning nonpropagation condition for Rayleigh waves in a clamped half-space, and Love's theorem that describes condition of nonexistence for Love waves. The numerical simulations reveal that to be effective the length of the horizontal barriers should be comparable to the typical wavelength.

Generally, current methods for preventing failure of structures due to seismic activity can be divided into two groups: (i) approaches for creating seismically stable structures and joints; this group contains different methods ensuring either active or passive protection; (ii) approaches for creating a kind of seismic barrier preventing seismic waves from transmitting wave energy into a protected region.

While the first group includes a lot of different engineering approaches and solutions, the second one contains very few studies; see Takahashi et al. [

The considered seismic barriers can be of two types: vertical, aimed to reflect, trap, and dissipate most of the seismic wave energy; horizontal, based on Chadwick and Smith [

Vertical and horizontal seismic barriers.

Yet another interesting approach is to create a “rough” surface of the half-space to force the propagating Rayleigh wave scatter by caves and swellings, see (Figure

Rough surface acting as a seismic barrier against Rayleigh waves.

In practice, such a rough surface can be achieved by a series of rather deep trenches oriented transversally to the most probable direction of the wave front. Some of the obvious deficiencies of this method are (i) its inability to persist the surface waves other than Rayleigh waves; (ii) protection from Rayleigh waves travelling only in directions that are almost orthogonal to orientation of the trenches; (iii) high sensitivity to the frequency of travelling Rayleigh waves. These shortcomings made an idea of exploiting a rough surface as a kind of protective barrier unrealizable.

Horizontal barriers can be constructed by modifying properties of the outer layer preventing the corresponding surface wave from propagation.

In practice, modifying physical properties of the outer layer can be achieved by reinforcing ground with piles or “soil nails”; see papers where reinforcing was studied in a context of increasing bearing load of soil [

If distance between piles is sufficiently smaller than the wave length, then a reinforced region can be considered as macroscopically homogeneous and either transversely isotropic or orthotropic depending on the arrangement of piles. Naturally, the homogenized physical properties of the reinforced medium will depend upon the material of piles, distance between them, and their arrangement.

For stochastically homogeneous arrangement of piles and the initially isotropic upper soil layer, the reinforced soil layer becomes transversely isotropic with the homogenized (effective) characteristics that can be evaluated by different methods.

Yields the upper bound for effective characteristics [

This method is related to constructing the homogenized inverse tensors

Much more accurate results yield the two-scale asymptotic expansion method [

Methods for constructing the corrector within the two-scale asymptotic expansion methods are discussed by Michel et al. [

In this section, we proceed to the analyzes of the main types of seismic surface waves and conditions for their nonexistence.

These waves discovered by Lord Rayleigh [

Rayleigh wave in a half-space.

One interesting problem associated with Rayleigh waves is the search of “forbidden” directions of “forbidden” (necessary anisotropic) materials that does not transmit a Rayleigh wave along some directions. Forbidden materials and forbidden directions have been intensively searched both experimentally and numerically [

Despite proof of the theorem of existence, a small chance for existence of forbidden materials remained. This corresponded to the case of nonsemisimple degeneracy of a special matrix associated with the first-order equation of motion; actually, this matrix is the Jacobian for the Hamiltonian formalism used for Rayleigh wave description. However, it was shown [

These waves were introduced by Stoneley [

Stoneley wave on the interface between two contacting half-spaces.

Love wave propagating on the interface.

In contrast to Rayleigh waves, Stoneley waves can propagate only if material constants of the contacting half-spaces satisfy special (very restrictive) conditions of existence. These conditions were studied by Chadwick and Borejko [

It should be noted that for the arbitrary anisotropy no

Love waves [

In the case of both

SH waves resemble Love waves in polarization but differ in the absence of the contacting half-space. At the outer surfaces of the layered plate, different boundary conditions can be formulated [

Besides Love and SH waves, a combination of them can also be considered. This corresponds to a horizontally polarized wave propagating in a layered system consisting of multiple layers contacting with a half-space. Analysis of conditions of propagation for such a system can be done by applying either transfer matrix method [

At present (2010), no closed analytical conditions of existence for the combined Love and SH waves propagating in anisotropic multilayered systems are known; however, these conditions can be obtained numerically by applying transfer or global matrix methods; see [

Different observations show that genuine Love and the combined Love-SH waves along with Rayleigh and Rayleigh-Lamb waves play the most important role in transforming seismic energy in earthquakes (e.g., [

Lamb waves [

More interesting from seismological point of view are Rayleigh-Lamb waves; see (Figure

Rayleigh-Lamb waves.

3D model of the horizontal round-shaped barrier interacting with a long Rayleigh wave: (a) 3D model; (b) cross-section near the barrier.

Herein, we present some results on numerical simulation of propagating seismic waves and their interaction with horizontal seismic barriers. The presented results were obtained by the explicit FE code.

Our analyses revealed that similarly to vertical barriers, the transverse barriers should satisfy several conditions to effectively protect from seismic waves: (i) length (horizontal) of the barrier should be comparable to the wavelength; (ii) material of the barrier should have larger density than the ambient soil for Rayleigh waves; that is, in agreement with Chadwick’s theorem stating that at the clamped surface of a halfspace, no Rayleigh wave can propagate; (iii) material of the barrier should satisfy the opposite Love’s propagating condition (

Figure

To simplify the subsequent analyses, a 2D model was used for numerical studies; (Figure

2D model with seismic barrier.

Seismic waves emanating from the epicenter.

In the center of the upper edge of a plate a harmonic loading was applied allowing us to model propagation of seismic wave. To analyze effectiveness of the barrier, the magnitude of displacements was evaluated at two points on the upper edge located at equal distances from the epicenter: one point in the left part of the plate and another in the right part behind the barrier.

Comparing both sides of a plate gives information on the effectiveness of the barrier in respect of the reduction of the magnitude in the right side of a plate behind the barrier.

In accordance with Buckingham's pi-theorem [

The preliminary analysis revealed that both Poisson's ratios have almost no influence on the displacements at

Variation of the magnitude of displacements at the reference point

Variation of the magnitude of displacements at a fixed point behind the barrier due to variation of

The charts in Figure

The most interesting and surprising deduction that can be made from analyzing these charts states that a relatively broad variation of Young's modulus of the barrier at the constant modulus of the half-space almost negligibly influences magnitude of displacements in the protected zone.

Variation of the magnitude of displacements at the reference point

Variation of the magnitude of displacements at a fixed point behind the barrier due to variation of

The charts in (Figure

Herein, the influence of the relative dimensions of the barrier

Figure

Variation of the magnitude of displacements at a fixed point behind the barrier due to variation of the relative length

Similarly, the dependence of the magnitude of displacements on the depth of the barrier is given on (Figure

Variation of the magnitude of displacements at a fixed point behind the barrier due to variation of the relative depth

As can be predictable, increasing either relative length or relative depth of the barrier can considerably reduce magnitude of vibrations in the protected zone.

Herein, a brief outline of future research directions related to creating more efficient seismic barriers is given. A practically important case, when seismic barriers appear to be indispensable, is discussed in the last subsection.

To make search of the optimal geometric and physical properties of the protecting barriers more systematic, solution of the following optimizing problem can be suggested. Mathematically, the optimization problem for minimizing magnitudes of deflections can be written as finding minimum of the following target function

Simple observations reveal that different types of seismic barriers can be most efficient at soft soils especially subjected to liquefaction, when more traditional seismic protection measures can be inadequate. Indeed, by diminishing amplitude of seismic waves inside the protected zone, the considered barriers should improve stability of liquefied soils.

However, for such soils, a more complicated analysis of traveling waves involving Biot’s theory of poroelasticity can be needed; see E. Detournay and Cheng [

The authors thank the Russian Foundation for Fundamental Research (Grant no. 09-01-12063-ofi) for partial financial support.