From all the above results it is obvious that the inversion process must not be guided solely by the comparison of the experimental dispersion curve and the dispersion curve for the theoretical first Rayleigh mode. Influence of higher Rayleigh modes should be considered through evaluation of the modal displacements, the rate of energy transmission or mode shapes.
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Figure 14. Normalized rate of energy transmission for the first four Rayleigh modes.
Two procedures at this moment seem to represent the simpliest approaches. The first one is based on the direct comparison of the experimental and the “simulated” dispersion curves. This approach provides a comparison of the experimental dispersion curve, in which Rayleigh waves are contaminated by body waves, with an equivalent
numerically simulated one. The second approach is based on the comparison of the experimental dispersion curve and the “average phase velocity” dispersion curve. The “average phase velocity” curve represents a weighted average of the phase velocity of several modes, in which the weighting factors are represented by modal displacements. The
“simulated” and the “average phase velocity” dispersion curves are compared to the theoretical dispersion curves for the plane Rayleigh waves in figure 15.
CONCLUSIONS
Conclusions from the study on Rayleigh wave dispersion in soil profiles where the shear wave velocity does not generally increase with depth are:
1) Higher Rayleigh modes provide a significant, and in many cases, a dominant influence on the overall wave
propagation pattern along the surface of a system. Therefore the inversion of the experimental dispersion curve should not be guided solely by the theoretical first Rayleigh mode.
2) Transition of influence from one Rayleigh mode to another is characterized either by localized approaches or by significant changes in curvature of the corresponding dispersion curves.
3) The presented results suggest that the inversion process should be based on the direct comparison of the experimental either with the “simulated” or with the “average phase velocity” dipersion curve.
Figure 15. Theoretical dispersion curves for the first four modes for plane Rayleigh waves, the “simulated” and the “average phase velocity” curve.
REFERENCES
Gucunski, N. (1991), Generation of Low Frequency Rayleigh Waves for the Spectral-Analysis-of-Surface-Waves Method, Ph.D. Dissertation, Department of Civil Engineering, The University of Michigan, Ann Arbor.
Gucunski, N. and Woods, R.D. (1991), “Use of Rayleigh Modes in Interpretation of SASW Test,” Proceedings of the Second International Conference on Recent Advances in Geotechnical Earthquake Engineering in Soil Dynamics, Vol. II, St. Louis, Missouri, March 11–15, pp. 1399–1408.
Heisey, J.S., Stokoe, K.H.II, Hudson, W.R., and Meyer, A.H. (1982), Determination of In Situ Shear Wave Velocities from Spectral Analysis of Surface Waves, Research Report No. 256–2, Center for Transportation Research, The University of Texas at Austin, December, 277 pp.
Nazarian, S. and Stokoe, K.H.II (1983), Evaluation of Moduli and Thicknesses of Pavement Systems by Spectral-
Analysis-of-Surface-Waves Method, Research Report No. 256–4, Center for Transportation Research, The University of Texas at Austin, December, 123 pp.
Nazarian, S. (1984), In Situ Determination of Elastic Moduli of Soil Deposits and Pavement Systems by Spectral- Analysis-of-Surface-Waves Method, Ph.D. Dissertation, Civil Engineering Department, The University of Texas at Austin.
Nazarian, S. and Stokoe, K.H.II (1986), “Use of Surface Waves in Pavement Evaluation,” Transportation Research Record, No. 1070, pp. 132–144.
Sanchez-Salinero, I., Roesset, J.M., Shao, K.-Y., Stokoe, K.H.II and Rix, G.J. (1987), “Analytical Evaluation of Variables Affecting Surface Wave Testing of Pavements,” Transportation Research Record, No. 1136, pp. 86–95.
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Transient Response of Certain Topographical Sites for SH-Wave Incidence
H.Takemiya, C.Y.Wang, A.Fujiwara
Department of Civil Engineering, Okayama University, Okayama, Japan
ABSTRACT
In view of the past seismic damages the soil amplification with modified predominant period is pointed out as a crucial factor. This paper is concerned with the analysis of the wave propagation and scattering through a certainly
topographical-shaped alluvium.The time domain boundary element method is applied with use of elementwise analytical double integrals over space and time domains. For the SH wave incidence, the time history response of canyon/ alluvium surface are computed and interpreted from the engineering viewpoint.
INTRODUCTION
Observations of past earthquake damages are very indicative of the seismic wave amplification by alluvium from the rock-like base level. The vertical shear wave propagation (1-D theory), has been dominantly employed for evaluating such surface soil amplification. In view of the surface/subsurface irregularities like a canyon or an alluvial valley, the 2- D modeling should be made at least to interpret the scattered wave field.
The steady state harmonic analyses for certain topographical site condition have extensively been conducted. The amplification/reduction due to such topographies in comparison with the far field without it has been investigated with respect to type of the incident wave, angle of the incidence, and nondimensional frequency such defined as the ratio between the width of the site and the incident wave length. This response characteristic gives a meaningful interpretation of the topographical effect on seismic waves. Frequency domain analysis was first started on the antiplane motion for its simplicity. The investigation on the in-plane motion was then followed by many researchers. The methodologies for these studies are a series of expansion by wave functions, the finite element method (FEM), the so-called Aki-Larner method which assumes the discrete wave number expansion, the boundary element method (BEM), and the hybrid method of the FEM and the BEM, depending the complexities of the topography (e.g., Takemiya [1]). Extensive reviews of the topic are seen in the works by Sanchez-Sesma, et al. [2], Mossessian and Dravinski[3].
Another important aspect is the response time history or the transient response, which gives the phase characteristic for wave propagation and scattering due to the presence of the surface/subsurface soil irregularities. Reviewing past works concerned, the finite difference technique was taken by Ohtsuki [4]; the discrete wave number boundary element method was developed by Kawase [5]. The Fourier synthesis method was the straightforward approach to get it from the above steady state solution through the Fast Fourier Transform (FFT) algorithm as Mossessian and Dravinski [6] showed. However, care should be taken for the phase variation since the FFT presumes a certain periodic duration. The direct time domain BEM analysis is a promising approach for transient response problems. Discretization both in space and time for the boundary integral equation needs the elementwise double singular integrals over these domains. The author succeeded in getting such an explicit solution (Wang and Takemiya [7]). The time stepping algorithm facilitates the transient response computation for a prescribed incident wave.
Herein, the out-of-plane motion of a canyon/alluvium on a uniform elastic halfplane base is analyzed for the SH wave incidence. The point of interest is placed on the wave scattering to be characterized by the phase by taking the Ricker wavelet as an incident wave function. Also, interested is the soil amplification for sinusoidal incident wave function to compute until the would-be steady state response is attained.
FORMULATION
Superposition of wave fields
The soil domain which includes a certain topographical alluvium is substructured into the surface soil deposits and the surrounding far field. The input seismic motion is prescribed as an incident wave to be defined at the far field (denoted by superscript F). The presence of the subsurface of soil deposits, reflecting the incident waves at the interface with the far field, generates scattered waves (denoted by the superscript S) in the far field. The near field is composed of the transmitted waves and the reflected waves from the free surface. See Fig. 1.
Fig. 1 Wave field for an alluvium desposit
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Free field motion
The SH wave incidence is considered. The displacement is prescribed by
uI(x,z,t)=a f(αI) H(αI) (1)
in which f(αI) denotes a certain function to describe the wave form. Herein, the Ricker wavelet of representative wave length λc=VsT in which Vs denotes the shear velocity and T is the representative period is used. Then,
uI(αI)=[2(πλcαI)2−1] exp{−(πλcαI)2} for Ricker wavelet (2.1) uI(αI)=1−cos(2πηαI) for sinusoidal wave (2.2) with dimensionless frequency η=2a/λ (2a=width of the soil deposit, λ=wave length). H() stands for the Heaviside step function and a is a constant. The argument αI indicates the phase to be specified at location (x0, z0) at time to as
αI=Vs(t−t0)−sinθ(x−x0)+cosθ(z−z0) (3)
The reflected wave at the free surface keeps the wave form but changes the phase
αR=Vs(t−t0)−sinθ(x−x0)−cosθ(z+z0) (4)
The total free field response is then
UF=uI+UR (5)
Boundary Integral Equation for Wave Scattering
The scattered wave propagation of the far field is computed by the BEM. The boundary integral equation representation in time domain for elastic waves scattering is formulated from the reciprocal theorem.
(6) in which u and t are the displacement and traction of the concerned body; u* and t* are the Green functions (or the fundamental solutions) for displacement and traction at field location y at time t due to a unit impulse force at location x at initial time equal to zero; the symbol (*) denotes the convolution integral operator with respect to time. For a 2- dimensional full space of an isotopic, homogeneous, elastic solid, the fundamental solution, when the out-of-plane motion is concerned, is given by
(7), (8) in which τ=Vst, r=|x-y|, n(y) is the outward normal of the boundary S(y) for the concerned domain. The integral p.v.∫( ) is interpreted by the Cauchy’s principal value and c(x)u(x) is the so-called free term.
Spatial and Temporal Discretization
The boundary is discretized into the E linear segments within each the (M+1)
nodes exist. The displacement and traction in the e-elment can be expressed approximately in terms of the nodal values with the aid of the assumed interpolation function, .
(9) in which ξ is the local coordinate within the element. Substituting this into Eq.(6) yields
(10) or in a matrix form
CU(t)=G(t)*T(t)−H(t)*U(t) (10)′
The time axis is divided into a sequence of equal increment ∆t so that the time tk= k∆t (k=1,.., K; K refers the current time). The time variation of response is also approximated by use of the interpolation function φNu for displacement and φNt for traction as
(11) Substitution of Eq.(11) into Eq.(10) results in
(12)
Time Stepping algorithm
Under the assumption of an identical interpolation function for every step, which spans ∆t only, Eq.(12) is rewritten as
(13) in which
(14), (15) with δk=1 for k=0 and δk=0 otherwise. Eq.(13) is solved stepwise for the unknown quantities based on the known quantities at previous times.
In the above formulation the crucial part lies in the execution of the elementwise double integral operation as defined by (16) which is singular about t* when xl is at the e-element. The analytical solution has
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been obtained for a combination of representative space and time interpolation functions by use of the Cagniard-de Hoop method (Wang and Takemiya, 1991). The simplest case is for the 0-th order element or the constant element whichleads
(17) For other elements the reader refer to the above publication.
Substructure formulation
For the wave field analysis for an alluvium on a uniform halfspace, the substructure procedure is effectively used. Referring to the illustration in Fig. 2, the separated alluvium deposits are characterized by the discretized form of the boundary equations.
(18) The exterior halfspace should be treated only for the scattering wave, not including waves as the free field, so that it is governed by
(19) Condensing out other variables than those related to their interface, the governing equations for the respective domain are expressed with the interface variables as unknown quantities.
(20), (21) The continuity condition is claimed to make an original total couped domains such that
(22), (23)
Fig. 2 Substructure formulation