In this section, we describe several different representations of the results of the 2D ground motion simulation, and compare the results with those from corresponding 1D models. We also introduce a simple rectangular valley to assist in the interpretation of the results, and calculate response spectra of simple structures located at different sites within the Kirovakan model in order to relate the simulated performance to the observed damage. In addition, a response spectrum is calculated for the city of Leninakan, as a means for explaining differences in the observed structural performance between the two cities.
Synthetic accelerograms for several points along the free surface of the valley and its vicinity are shown in Fig. 5a. The locations of these points are identified by the labels A to I in Fig. 2. For points outside the valley the motion differs little from the free-field motion. Inside the valley, however, the response is amplified significantly, especially in Zone 2. This can further be seen in Fig. 5b, in which the PGA along the entire valley surface is graphed as a solid line. In Zone 3, the average amplification with respect to the free-field value is about 30 percent, with a slight increase toward the right edge, which is underlain by the softer soils. The site effects are most pronounced within Zone 2, in which the PGA exceeds 0.5 g at a number of locations. It is also noteworthy that the peak response is highly oscillatory across this region. Such rapid spatial variation of the ground motion has actually been observed in real earthquakes, e.g., during several aftershocks at clusters of sites that were instrumented in the aftermath of the 1994 Northridge earthquake (Hartzell et al, 1996; Bardet and Davis, 1996). This behavior can have important practical implications; yet, it is generally not possible to reproduce it via 1D simulations. To illustrate this point, and to help gain a better understanding of the differences between 1D and 2D effects on site response, a 1D simulation was also conducted for the idealized Kirovakan valley under consideration. For each mesh point on the valley surface a 1D analysis was performed for a soil column whose properties are identical to those beneath that point using as input the same incident SH-wave as for the 2D simulation. The distribution of the 1D PGA across the valley is also shown in Fig. 5b, by a dashed line. In contrast to the 2D simulations, the peak response is constant along sections where the valley bottom is flat, and varies only gradually as the valley depth changes, except near the edges. Overall, in Zone 2 the PGA is significantly greater for the 2D than for the 1D simulations.
In order to help explain the rapidly oscillatory nature of the PGA across the valley, it is convenient to filter temporarily the effect of the earthquake excitation and concentrate on the valley response to a simple steady-state harmonic excitation. The results are best observed in the frequency domain. Figure 6 shows the Fourier spectral ratio (FSR) of the response at site F near the point of maximum depth (shown in insert), as a function of frequency, corresponding to both the 2D and 1D simulations. This ratio is obtained by dividing the amplitude of the Fourier transform of the synthetic accelerogram at F by twice the amplitude of the Fourier transform of the free-field accelerogram (Fig. 4b). The two lines shown on Fig. 6 also represent the 1D and 2D amplification ratios of the response at F with respect to the free-field motion on the surface of a halfspace made up of the same bedrock material as that underlying the valley model (Fig. 2), due to a steady-state harmonic vertically incident SH-wave. It is apparent from this figure that: (1) The 1D amplification ratio exhibits the resonant behavior typical of flat-layered systems. For this particular example the largest peak corresponds to the second resonance, in the vicinity of 2.5 Hz; (2) The 2D amplification ratio also exhibits resonant behavior in the vicinity of the 1D resonant frequencies, but the corresponding 2D frequencies are slightly higher than those for the 1D results, due to the lateral confinement of the valley. The values of the respective peaks, however, are considerably larger for the 2D case; (3) In addition to the essentially 1D resonant frequencies, the 2D valley experiences resonant behavior at other frequencies, which appear to be unrelated to the 1D case. The amplification ratio oscillates rapidly with frequency, reaching peak values that greatly exceed the 1D values. Interestingly, for certain frequencies the 2D amplification ratio is much smaller than unity, denoting, in effect, a strong deamplification, or destructive interference of seismic waves.
The Fourier spectral ratios shown in Fig. 6 are for a single point on the valley surface. To examine how the FSR varies from point to point, one can construct similar spectral curves for all the surface mesh points across the valley and plot the results as contour amplification ratios in terms of both frequency and location. These contours are shown on Fig. 7 for the 2D and 1D simulations, thus revealing simultaneously the spatial and frequency distribution of the ground motion. The scale of the FSR is given by the color bar. Similar contour displays have been presented by Sánchez-Sesma et al (1993) for homogeneous valleys of simple geometrical shapes. For the frequency range considered in Fig. 7, the 1D simulations exhibit four resonant frequencies near the deepest part of the softer subvalley. As expected, the values of these frequencies increase away from the center as the valley becomes shallower. Within the stiffer Zone 3 there is only one resonant frequency. The results of the 2D simulation are much more complex. First, multiple resonant frequencies occur throughout the valley. For some of these frequencies the amplification ratio reaches a maximum value of 8, almost double that for the 1D case. For each resonant frequency the amplification ratio exhibits several peaks, whose number increases with frequency. Between the peaks, the amplification ratio almost vanishes at certain locations. This means that the ground surface essentially remains at rest for these frequencies and locations, and experiences strong motion at nearby points. By contrast, for the 1D case the FSR varies only gradually across the valley.
To further elucidate these features of the response of the 2D valley, we consider next a simple example involving a rectangular, undamped, homogeneous valley of width L, depth H, supported on a rigid base and rigid side walls. While this model is substantially simpler than the valley under study, its dynamic behavior will prove to be useful for the interpretation of Fig. 7. Bard and Bouchon (1985) have used the same example for a study that focused on the fundamental resonant response.
The natural frequencies of the valley, when subjected to antiplane vibrations, can be written as:
in which 
is the fundamental
natural frequency that the valley would exhibit if its width were
infinite; i.e., 
is the natural frequency of a flat layer
on a rigid base. Its value is given by
, in which 
is the shear
wave velocity of the medium.
By setting the origin of a right-handed cartesian coordinate
system at the lower left corner of the valley, the horizontal x-axis
along the width, and the vertical y-axis toward the free surface, the
mode shapes 
, associated with the natural frequency
, are given by:
The integer m denotes mode shapes in the heightwise direction and n
along the width. The first factor on the right side of Eq. 6
corresponds precisely to the mode shapes of the flat layer, whereas the
second factor introduces the effect of the finite width. Thus, for
each mode shape of the flat layer (i.e. for a fixed m), there is a set
of mode shapes in the widthwise direction with corresponding frequencies
. Along the width the mode shape 
exhibits crests
or troughs at a set of points 
with abscissas
while points 
at
are nodes and remain at rest.
The locations of the points 
and 
are
shown on Fig. 8a together with their corresponding
natural frequencies, for m = 1 and a particular valley shape
ratio ( 
). The corresponding points for m = 2 (i.e. those
associated with the second mode shape in the heightwise direction) are
shown in Fig. 8b. The
symbols, asterisks and circles, denote, respectively, extremum
(crests and troughs) and
nodal points. Thus, for instance, the mode shape 
has a natural
frequency 
, a single extremum at
and
no nodes; the natural frequency associated with the mode shape 
,
which has two extremum values, at 
and 0.75, and a nodal
point at 
, is 
; similarly,
the mode shapes 
and 
have natural frequencies
and 
, and
the same extremum and nodal points as 
and 
; etc.
One can expect that if the valley is excited at a prescribed
frequency 
, those modes associated with the natural frequencies
which are closest to 
will be the ones that
respond most strongly. The amplitude of the surface response will
thus be greatest at points located near the critical points 
and smallest in the vicinity of the nodal points 
.
The top plot in Fig. 7a shows that the idealized Kirovakan valley also tends to respond in its modal shapes, exhibiting crests and troughs, and nodal points, just like the much simpler rectangular valley, even though the Kirovakan model does not have separable mode shapes along its height and width. In fact, if one now examines the lines joining the extremum points and the nodal points, as shown in Fig. 8, the analogy between the two situations becomes apparent. We call the two sets of lines, extremum and nodal lines, respectively. From a comparison of Figs. 7 and 8, it is seen that the amplification ratio contours have a skeleton of extremum and nodal lines in the space-frequency domain and that the largest crests and troughs occur at distinct resonant frequencies akin to the natural frequencies of the simple rectangular valley. These resonant frequencies are given by the imaginary parts of the complex eigenvalues of the homogeneous problem associated with Eq. 2. The crests, troughs, and nodes are clustered along bands rather than lines, due to the presence of damping.
Notice that for the particular example of Kirovakan considered herein the larger amplification ratios do not correspond to the fundamental frequency, but occur at higher ones, with a clear 2D effect. It is also noteworthy that the motion in Zone 3 is influenced by that in Zone 2, by the energy that leaks through their interface; one can observe how the extremum and nodal lines generated within Zone 2 extend to Zone 3, although the amplitude of the response decreases drastically in the stiffer zone. None of these effects is present in the 1D simulation, as shown at the bottom plot of Fig. 7.
We return now to the earthquake problem in order to examine
how the valley ground motion affects the response of structures located at
different points within the valley. To this end, we consider next the
pseudo-acceleration response spectra shown in Fig. 9a. In
this figure,
the maximum pseudo-acceleration 
, in
which 
maximum story drift
of a single-story structure with an undamped natural frequency 
.
We recall that the maximum base shear, V, in the structure is related
approximately to A through
, in which m = mass of single-story structure. That is,
is the seismic coefficient by which the weight must be
multiplied to obtain the maximum base shear.
Jennings (1997) has shown
that the base shear force of a regular n-story structure,
, can also be expressed approximately in terms of 
for a single-story, as 
, in which M is the
total mass of the superstructure. Thus, the results of Fig. 9 can also
be applied to regular multi-story buildings. The spectra in Fig. 9a
were calculated for structures located at three different sites P, Q, R,
identified in the insert, using as excitation the synthetic accelerograms
obtained at those points from the 1D and 2D simulations of the 1988
Armenia Earthquake. Soil-structure
interaction is not taken into consideration. With an average shear
wave velocity of 280 m/s over the top 30 m of soil in Zone 2, this
effect is not expected to be significant for the structures and
frequency range under study. Damping in the structures is 5 percent
critical. Dashed and solid lines are used for the 1D and 2D results,
respectively, and the suffix K in the key refers to Kirovakan.
At points P (outside the valley) and Q (in Zone 3), the response spectra
corresponding to the 1D and 2D simulations essentially coincide with
each other. This indicates negligible 2D effects at these points.
On the other hand, the 1D and 2D spectra
for point R in the middle of Zone 2 show a huge discrepancy: the
pseudo-acceleration for the 2D ground motion exceeds the corresponding
results of the 1D simulation by as much as 100 percent. This clearly
illustrates how 2D site effects can sometimes have a dramatic effect on
the earthquake response of structures.
At this point it is of interest to compare the response spectra obtained for Zone 2 in Kirovakan with one derived for Leninakan, due to the very different levels of damage experienced in the two regions. The dashed-dotted line labeled L on Fig. 9a is the response spectrum calculated with a base excitation obtained from a 1D simulation using the soil properties for Leninakan reported by Yegian et al (1994b). The corresponding soil column was subjected to a vertically incident SH-wave with the reference rock motion scaled to 0.25 g, the estimated PGA at the rock outcrop in the Leninakan region. Since this city is located in a large and shallow valley, a 1D analysis seems sufficient for determining the soil amplification effects. Building-damage statistics indicate that 62 percent of the four- to five-story structures in Zone 2 in Kirovakan, with frequencies from 2.5 to 4 Hz, collapsed, whereas only 21 percent suffered the same fate in Leninakan. With this damage distribution one would expect that the response spectrum for Zone 2 should be significantly higher than the corresponding one for Leninakan. However, the computed spectra for L and R based on 1D ground motion simulations (Fig. 9a) are quite close to each other, especially for frequencies greater that 2.5 Hz. If, on the other hand, one compares the Leninakan spectrum with the solid line spectrum for R, corresponding to the 2D simulation in Kirovakan, it becomes clear that the seismic forces generated during the 1988 Armenia Earthquake must have been much stronger in Zone 2 in Kirovakan than in Leninakan. This is in agreement with the observed damage, and suggests that in order to explain the structural behavior within the Kirovakan Valley it is essential to take its finite lateral extent into consideration.
To gain further insight into the response of structures within Zones 2 and 3 in Kirovakan, the pseudo-acceleration spectra for several natural frequencies are shown in Figs. 9b to 9d for all points across the valley. These figures again illustrate that 2D site effects, in general, increase significantly the structural response over that from a 1D analysis. Moreover, the rapid spatial variation exhibited by the ground motion amplification is also observed in the pseudo-acceleration spectra. This is of considerable practical importance, as it means that two identical structures separated by a short distance can be subjected to widely different seismic forces even if the underlying soils have essentially the same properties.
In order to examine simultaneously how the peak structural response caused by the valley ground motion varies with the structure's natural frequency and its location within the valley, a novel representation of the response in the form of space-frequency pseudo-acceleration spectra is introduced here. Results are presented in Fig. 10 for structures with 5 percent critical damping for the base excitations obtained from the 1D and 2D simulations. The natural frequency of the structure is given on the abscissa and the location within the valley is shown on the ordinate. The corresponding value of A/g is given by the color bar. Several observations can be made from this figure:
1. Structural response in Zone 3 is small compared to that in Zone 2. The strongest shaking occurs in the middle of Zone 2 and near its confluences with Zone 3 and the rock outcrop, for frequencies in the range of 2 to 5 Hz. These are precisely the locations within the valley and the natural frequency range for which damage was strongest during the 1988 earthquake.
2. The strongest response by far, however, corresponds to structures with natural frequencies close to 2.5 Hz. This is a classical double resonance phenomenon. Under the incident seismic wave, the valley responds in its most sensitive resonant mode shapes, whose frequencies almost happen to coincide with the dominant frequency (2.5 Hz) of the earthquake excitation (Fig. 4b). The second resonance occurs when the amplified ground motion produces strong vibration of structures whose natural frequencies coincide with those being excited in the soil deposits.
3. The 1D response spectra also predict resonant behavior near 2.5 Hz. The spectral ordinates, however, are much smaller than those for the 2D ground excitation. Also, whereas the 1D spectra are quite diffuse, the 2D spectra define sharp, localized regions, both in space and in frequency, for which the response is strongest. Overall, the spectral ordinates are significantly greater for the 2D case.
4. The 2D spectra exhibit islands of strong response in the 3 to 5 Hz region. The 1D spectra show no such behavior.
5. The nodal and extremum lines identified in the Fourier Soil Response Spectra in Fig. 7 are also apparent in the structural response spectra. While this response is largest within a region near the deepest portion of the valley, there is an adjoining region which essentially follows a nodal line. Thus, a small change in the structure's location can result in a dramatic change of its response. In addition, since the response spectra also vary rapidly with frequency, this means that two slightly different structures located essentially at the same site can experience substantially different responses.
