CONCLUSIONS AND RESULTS

Stage 1

Using experimental curves obtained by dynamical laboratory tests, analytical expressions for the elastic modulus and attenuation were calculated as functions of the deformation level. Pre-existing state of stress (or deformation) was also considered. To accomplish these experimental performances extant geotechnical data base of soils characteristics was used.
The experimental data obtained by these studies are useful in describing nonlinear viscoelastic behavior under dynamic stress. It can be observed a major influence of deformation level upon shear modulus and damping (Fig. 1a, b) and a small one, of over 1 Hz frequency, on frequency functions Gk(ω) si Dk(ω).

Nonlinear behavior of the soils under cyclic stress (like seismic conditions) is emphasized. At highest stress level, flow phenomenon and energy absorption may occur.
Geotechnical tests performed on samples obtained by drillings have pointed out nonlinear behavior induced when a certain strain level is exceeded. For example, the laboratory tests performed on Hardin and Drnevich resonant columns have shown consistently a decrease of the dynamic modulus function G and an increase of the damping function D while the strain γ goes over 10^-4 %, G=G(γ), D=D(γ).
The decreasing of normalized values for the torsion modulus G_n = G/G₀ during the experiments is smaller for hard materials as granite or marly loess, while soft materials, which are found in superficial layers, have high variations of G_n and damping D_n = D/D₀, for the same strain domain.

Stage 2

Establishment of the mechanical model according to soils geophysical properties involved in seismic wave propagation, and their surface emergence. Typical laws were elaborated with physical parameters corresponding to deep and surface soils. For the latest we are focused on critical situation inducing a high risk. Physical weights with nonlinear features were introduced, which describe more appropriately the surface layers behavior. This way it was highlighted the effect of geological strata situated on seismic waves path, which hit afterward dense populated areas.
Developing a specific computational procedure consisting of approximation of the general equation solutions of the medium by series expansion in powers of a small parameter which includes as a novelty the above specified additional weights. Linear equation systems are obtained for each approximation step.
Development of intensity determination procedure of the seismic processes induced by reflection and refraction, superposition effects, for stratified, foliated media. There have been selected in our study certain computational data corresponding to urban areas in which weak consolidated soils are found that induce high risk caused by nonlinear amplification effects. These procedures are applied to the above mentioned equation.
Complete parameters set which define behavior of the seismic waves in complex stratified media and at the surface, taking into account nonlinear phenomena. By this procedure a thorough characterization is obtained, with the advantage of performing a seismic risk assessment in a much realistic manner.
Based on quantitative results, we are able to say that maximal seismic amplification for Bucharest region is 30%, amplification increasing by nonlinear effects, compared to linear ones, especially due to visco-plastic irreversible processes.
The results are included in tables and databases containing geological data for depth structures and mechanical parameters useful in local seismic effect determination at micro-local scale (widely presented in extended report).
Introduced, for emphasizing nonlinear effects, spectral amplification factor (SAF), as a ratio of the maximum spectral acceleration (Sa), velocity (Sv), relative displacement (Sd) to maximum values of acceleration, velocities, respectively displacement, from processed recorded accelerograms. It was emphasized a strong nonlinear dependence of SAF factors with the magnitude of strong earthquakes. New methods of studying the influence of diffraction of seismic waves upon stratified structures of soils and lateral discontinuities of propagation structures are showed.
Application of the methods of seismic movements simulation by structural models with well-known parameters, which have been a start point for acquiring knowledge concerning the method limitation and provided valuable parametric tests on the effect of the Vrancea seismic sources on the parameters of the seismic motion at the surface of the sedimentary sites. Construction of structural models to be employed in simulation means more than a simple series of in-situ determined parameters, or taken from the literature, it aims also at their application, depending on the possibilities and characteristics of the methods, the computation programs etc. This has been achieved at this stage by using specific computing codes.

Stage 3

Displacement, velocities and accelerations amplification were computed, for a resonance proximity, the attenuation parameter being instrumental in SAF definition, related to movement under external perturbation (disruptive) force exclusively. Amplification factors increase monotonically in time, being asymptotically limited, immediately estimated by semi-analytical means.
General solution herein obtained, for the attenuated harmonic linear oscillator, gives the opportunity of a spectral approach to seismic response, resonance regime, or resonance vicinity, being described in a particular form by the harmonics and external oscillations, if external force is time-periodic.
The treatment of the perturbation external shocks, a limit case in theoretical studies, but quite frequently in practice. The amplification is in this case insignificant for displacements, relative normal for velocities, and quite appreciable for accelerations.
Theoretical estimation for the amplification factors herein employed by this approach, are in qualitative agreement to empirical SAF derived for local maximum displacements, velocities and acceleration from recordings at different location of interest. Studying these latest ones their strong dependence was revealed on the location and magnitude, generating local effects depending on nonlinear features.
Conditions causing nonlinear effects were investigated for the strong displacements recorded during the last strong earthquakes (Vrancea epicentral area). These spectral amplification factors were used in order to have a quantitative expression for the nonlinear behavior of the soils and their nonlinear response.
Highlighted strong nonlinear dependence between SAF and magnitude, and for example SAF coming from the recordings at Bacau seismic station for the last three strong events were compared (august 30, 1986, may 30, 1990, may 31, 1990).
The evidence for amplification factors being decreased with strain is depicted in Figures (from the extended report, together with response spectra for the last century’s three major earthquakes) and in tables (presented in extended report), confirming also the fact that there is a tendency of acceleration to decrease while magnitude increases (when nonlinearity is acting). The theory of spectral amplification factors introduced herein is consistently to the results obtained in experiments with resonant columns and displacements maximum in both linear and nonlinear cases are controlled by frequencies de-amplified by nonlinearity.
In the situation when the wave front is characterized by high frequencies and long path time caused by altered structure (because of high strains) of the propagation medium, or superposition of the radiated waves from multiple sources, the seismic risk may be increasing by 30%. This situation was encountered during March 4, 1977 event (Mw=7.4) that has consisted in three successive shocks along a relatively long fault. Standard procedure in seismic risk assessment doesn’t take into account this instantaneous and special effects.
Concluding that major Vrancea earthquakes, by their particularities, may induce amplification caused by nonlinearities, in typical seismic effects produced by directivity features. So it may encounter the possibility that seismic risk to be sub-evaluated because of disregarding the nonlinear effects of third order (not discussed in this study), or reflection and refraction of seismic waves.

Fig. 2 Response spectrum for August 1986 (Mw=7.1) and May 1990 (Mw=6.4) Vrancea events, at Magurele-Bucuresti, with recorded and computed maximum accelerations, (amax) and (Samax), for different attenuation parameter (%). Spectral amplification factor (SAF) function of perioad also depicted. C coefficient is a measure of nonlinear effects.

Fig. 3 Response spectra for Bucharest-INCERC location, for Vrancea, August 1986 event (Mw=7.1), at 5% damping. Local structure is modeled viscoelastic nonlinear (left)and anelastic linear (right). Local response (amplification) is shaded.

Fig. 4. Nonlinear dependence of spectral amplification factors on Vrancea earthquakes magnitude.

Hybrid method

For local seismic effects evaluation was used hybrid method which has been conceived especially for bidimensional structures containing lateral inhomogeneities. This method is a combination of analytical technique of modal vibration summation for a bedrock geological structure through seismic signal is propagating from the source to geological profile, and finite differences technique for local geological structure (of a city), in which are included lateral inhomogeneities (2D analysis). The simulation results consist in complete temporal series of displacements, velocities and accelerations at surface. These seismograms could be modeled as the real ones, for obtaining response spectra, Fourier spectra, local amplification spectra, design spectra etc.

it may be concluded that local structure has the same amplification effect on the seismic signal for the radial and transversal components, with mean values of 1.2-1.5 for Mw<7 (VR901,VR902, VR04) events. The general behavior for the vertical component of the simulated seismic movement shows higher amplifications, Amax(2D)/Amax(1D) = 2.6, and 2.1 for VR902 and VR04, These amplifications are rapidly decreasing along the section until subunits values are reached, indicating the attenuation of the seismic signal. For magnitudes Mw>7 the amplification for the simulated Vrancea August 30, 1986 event for the radial and transversal components are a little bigger, with 1.8-1.9 values, while for the vertical component it reached the lowest value.
the amplifications (their values and spatial distribution) of the vertical seismic signal depend less on earthquake magnitude and much on triggering mechanism and hypocentral depths, as it may be seen from parametrical study of the hypocentral depth influence.
it can be stated that relatively spectral response of the local structure on radial and vertical components is very sensitive (different) to changing in focal mechanism, while for the the transversal component is relatively stable. The shifting of the spectra maximum values from 0.4 Hz to 0.9 Hz for RAD component, and from 0.6 to 0.9 Hz for VER component are indicating the fact that frequencies content of the seismic displacement are changing with magnitude or triggering mechanism.
hybrid method employed here has obvious performances in local effect evaluation and, eventually in local seismic hazard assessment (microzonation), starting from synthetic accelerograms simulation (temporal series of displacements, velocities, accelerations), taking into account sources characteristics (temporal dimension, seismic moment, localization, fault plane solution), lateral inhomogeneities of geological structures, and physical-mechanical characteristics of the local structures. The seismograms by this approach are credible and complete, in the meaning of containing all the phases of the volume and surface propagating waves in the specified frequency domain, through geological structures.

MS-SH method

the computation of the seismic input through MS technique includes the seismic source parameters (localization, depth, fault plane solution) and also geological structures features, through the seismic signal is propagating up to emplacement, and where lateral discontinuities could be modeled, for long epicentral distances, as Vrancea area – Bucharest is the case. By considering all these variables, the seismic input of the MS-SH method are surpassing the limitation, in fact overrating, coming from the common elastic linear halfspace, used for bedrock model in the classic SH approaches
a self-imposed condition in this approach consists in spectral completeness of the computed seismic signal. This condition is necessary because SH techniques preserve frequencies content changeless, modifying only amplitudes and signal duration computed at surface.
very important in seismic response evaluation for all SH techniques is choosing of the bedrock level, where seismic excitation is placed and then modeled up through stack of layers. The effects induced by transversal seismic waves through bedrock-surface distance are especially because of nonlinear dependence of shear modulus and damping on strain value for each layer, as it may be seen from previous stages results. From this test follows:

As a general conclusion, we may state that seismic input as well as its implementation level (depth) is of decisive importance in seismic response and local seismic effects evaluation.
for the vertical propagation approach the SH techniques may employ several mathematical models for soils, in linear equivalent approximation (SHAKE, EERA, ProShake) or in nonlinear hysteretic modeling (NERA, Tess etc). The MS-SH method herein employed is using nonlinear viscolelastic model because of its suitable and completeness feature in physical phenomena description induced by strong earthquakes in the superficial layers. By using this approach for uniform layers of loess, clays, sandy gravel and marls is highlighted nonlinear dependence for the effective values of the dynamical shear modulus and damping functions of the magnitude for intermediary earthquakes. These dependences are depending of analyzed material nature (soils type) and inducing local seismic effects, observed or evaluated by MS-SH method. The study of soils dynamical parameters dependence of earthquakes magnitude emphasizes the fact that at Mw≥6 magnitudes the nonlinear behavior of the soil must be considered for seismic movement parameter evaluation at free surface, that is local seismic hazard evaluation, especially for thick layers (tens of meters), and applying a nonlinear viscoelastic model is needed.

the validation of this method was made by comparing response spectra of recorded transverse acceleration to the simulated surface acceleration one, for certain events. The models that have obtained less errors (for example 7.438% for an emplacement in Bucharest) were used further in predictive evaluation for surface movement and for structure typically effects.

a decisive advantage of this method over other deterministic methods of local response and local effects evaluation is its application and validation at a quite satisfactory level regarding engineering seismology, for intermediary focal depths (60-180km) and up to 2 Hz frequency domain (0.5-1s). Usually computations are made for Bucharest city, where we find high buildings, over 10 storeys, with 0.05-1 Hz frequencies interval (long period, 1-20 s). Though, for practical purposes, for example old and not so tall buildings that exist big cities (including Bucharest), up to 10 storeys high, having natural oscillating period up to 1s (>1Hz), an extension of frequency domain is needed. Anyway, from numerical simulation and recordings could be observed that spectral maximum response of the terrain is shifting toward frequencies higher than 1 Hz (short periods) when magnitude is diminished. In consequence, seismic microzonation studies have to use a larger frequency domain. For particularly case of Bucharest city, exposed to intermediary Vrancea earthquakes, we consider the necessity of extension of frequency domain up to 3 Hz.

Stage 4

new matricial method for quasi-classic approximation is derived by considering infinitesimal division of the inhomogeneous medium;
the inclusion of high order effects in dispersion study for this method;
highlighted the effect of an inhomogeneous medium in wave propagation, or of the defects, localized, extended, spatial correlated;
the method offers the opportunity to systematically introducing wave-type correction to the geometrical rays approximation, by enlarging the capability to applying more and new mathematical approaches to the wave propagation-related phenomena.