influence of local geology on earthquake ground motion

The primary goal of this thesis is to study the influence of the local geology on the incoming seismic waves. One model is an exact analogue of the original layered system when no viscosity is present.

Several well-known linear viscoelastic models are commonly used for theoretical descriptions of the mechanism of energy dissipation in viscous media. If CY is the shear stress and E is the shear stress, then the stress-strain law can be written.

Fig. 2.1. THREE COMMONLY USED LINEAR VISCO- VISCO-ELASTIC MODELS

FREE SURFACE

NUMERICAL EXAMPLES

EXA CT TRANSIENT ANALYSIS OF A NONVISCOUS LAYERED SYSTEM
MODIFIED SHEAR BEAM MODELS FOR APPROXIMATE TRANSIENT ANALYSIS OF A LAYERED SYSTEM

Also, the total thickness of the intermediate layers between the top layer and the foundation is 550 feet for all systems. A list of the symbols to be used in this chapter is given below, and they are redefined when they first appear in the text.

The response of the _N th layer, uN(zN, t), to the input 2y(t) during the time the transmission takes place over the base interface, can be written as. 2, (Tu')N is the coefficient of upward transmission of the model at the base interface. To obtain the coefficient of upward reflection of the model, let a signal propagating downward within the Nth layer be f(t-zN/cN).

Therefore~ for steady-state excitation, Eq. 7·and 4.·9 together prove that the model and the multilayer system are completely analogous to each other, since they have the same transfer function. In the next section, two methods for the approximate analysis of the previous model are presented. For most engineering problems, it would be sufficiently accurate to consider only the first S modal equations (S :::: R) and thus some savings in computing time can be achieved.

The accuracy of the lump model is proportional to the number of lump masses, R, and since this number is limited by practical considerations, it is important to develop the following method which directly analyzes the continuous slider. A proof is given in Appendix I for the orthogonality of the modal shape functions on_ the interval from z.

Fig. 4.. 1 T ~~:E FIRS'I ' s~~.i. .EAR ,. :;:;AM MOD~L

Keff)S uN(O,t)

The Scan number should then be selected depending on the desired accuracy of the approximate solution. H.(w) and is defined as the amplitude transfer function. to compare with the exact solutions of the corresponding functions. If a large hump observed in the output spectra is associated with a prominent resonance of the layered system, this hump will be observable in both the undamped and the damped spectra.

It can be associated with strong frequency components of the input motion, which also cause a hump in the input spectra. 1 is undoubtedly related to the resonance in the fundamental mode of the layered system, and the other. The undamped spectral ratio fluctuates around AMP(w) and is below AMP(w) around any natural.

In general, the spectral ratio associated with greater spectral attenuation is around any natural frequency in the system. In other words, the oscillatory nature of the spectral ratio decreases with increasing spectral attenuation.

Fig. 4. 5 AMPLITUDE TRANSFER FUNCTION OF THE NONVISCOUS 4-LA YER SYSTEM

AMP{W)~,

To find the approximate transfer function, · let us assume a steady-state input y(t) = e iwt • The stability response at the base can be written. The quadruple-layer system used to demonstrate the applicability of the first model will be used here with the exception that the layered media are now assumed to be standard linear viscous solids. 74 then gives the values of IHI, N(w) I evaluated at the first 13 natural frequencies, which are listed in Table 4.

The inputs used in the calculation are the functions (y) a and (y) b which _are .sh.own.in.Fi,g. Although exact solutions for the surface response are not available for comparison, the approximate solutions can be expected to have sufficient accuracy comparable to that obtained for the approximate transfer function. Damping of high frequency components is evident when the surface response is compared to that obtained for the non-viscous layered system.

On the other hand, the base response deviates only very slightly from that calculated for the non-viscous layered system, leading to the conclusion that the base motion is not as sensitive to pres-'. 5 seconds, the spectra deviate only slightly from the corresponding spectra calculated for the non-viscous system.

Table 4. 2 MODAL DAMPINGS OF THE 4-LAYER SYSTEM COMPOSED OF STANDARD LINEAR VISCOUS SOLIDS

AMP("')

Finally, it should be pointed out that in approximate analysis of the first model for a viscous layered system a. The reason for this is that the higher modes are damped in such a way that their contributions to the transfer functions in the prescribed frequency range are practically negligible. The allowance for using a smaller nurnber S implies some savings of the computing time required for the approximate analysis.

In terms of transfer function, this means that fHN(w)I is the constant I . approaches when the impedance ratio aN decreases. This observation suggests that, if N is small, a second model can be established as a further approximation of a given layered system by determining that the base motion is identical to the input motion 2y(t). The model is obtained by replacing the dashpot D in the first model by a rigid one to provide the identity between the base motion and the input, while the energy lost in the foundation due to its deformability is approximately accounted for by artificially an appropriate amount of viscous damping in the slider.

The S-mode representation of this model in terms of transformed normal coordinates, s {t), is shown in Fig. Consequently, unless N is effectively zero in a given system, there is always some damping in the model, even if the given system is not viscous.

Ceff)r= 2 f3rOJr (meff}r

Ceff}I

Keff) S

APPLICATIONS A. Introduction

The applicability of the theory thus depends on the extent to which the properties of the local geology and the arriving seismic waves deviate from an idealized layered system. However, if the seismic waves produce sufficiently large deformation of the subsoil, the non-linear, hysteretic behavior can be significant. The theory will therefore only be applicable if the horizontal dimensions of the subsoil are large compared to the depth.

For deep-focus earthquakes, especially those with a fairly short epicentral distance, the body waves will impinge almost vertically at the base of the subsurface. However, for destructive earthquakes with a shallow focus, the nature of the seismic waves is much more complicated. In this case, ground motion may be the result of waves arriving along different paths, and it appears that the subsurface response must be.

For shallow-focus earthquakes at greater distances (50 to 100 kilometers), the seismic waves will approach the local subsurface mainly in the horizontal direction, and the response of the local subsurface can only be approximately estimated by theory if the following wavelength criterion is met met . In the case where the body waves are incident obliquely, Haskell's matrix method can be applied. l 4, l S) Briefly, Haskell's method deals with layering problems exposed to obliquely incident plane body waves, and the response of the layered system is calculated in the frequency domain.

B EDROCK

L STATION

When the authors considered the crustal layers with a total thickness of approx. 10 kilometers, oblique incidence of the P waves was assumed and Haskell's method was used for a theoretical analysis. The difficulty came mainly from the propagation of the waves during the short period in the crust and the upper n1antle below the sediments. Also a diving of the crustal boundary of approx. 15 degrees was required to explain an azimuth deviation observed in the experimental results.

To find the condition under which important. subsurface resonance, let's first examine some examples of actual recorded earthquakes. The epicenter was along the San Jacinto fault zone in southern California, and San Onofre is along the Pacific coast, about 85 miles west of the epicenter. A comparison of the velocity spectra for the corresponding components shows no similarity in overall spectral features, nor coincidence in the locations of the spectral peaks.

Two possible reasons are given below to explain the apparent suppression of the local geological effect. That is, either because the soil layers are not well defined and the nature of the incoming waves was so complicated that resonance of the earth_ according to the theory did not occur, or.

SAN ONOFRE

Furthermore, he concluded that the dominant period of 2.5 seconds observed in ground motion is the fundamental natural period of the clay layers. 1 (a), we can expect the response of the first mode to be relatively large. The following is a medical example that demonstrates the dominant response of the first node as suggested by the theory.

2 = 7.53 rad/sec. It can also be shown that the modal participating factors for calculating the surface response of the model are The general characteristics of the output spectra are very similar to those of the spectra shown in Fig. MEXICO CITY. achieved considerable success in predicting the general characteristics of the velocity spectra of future earthquakes recorded in Mexico City.

A layered model must be established for theoretical interpretation of the recorded horizontal ground movements. This observation is confirmed by the small amplitude of the ground motion recorded in the peat (see Fig. Such anomalies could be partly contributed by the dispersion of the high-frequency components of the incident waves.

The second example demonstrated the persistent influence of soft subsoils in Mexico City on corded ground motions.

Fig. 5. 4 VELOCITY SPECTRA FOR THE SAN ONOFRE SHOCKS (N 57 °W)

APPENDIX II