Wave propagation in a seven-story reinforced concrete building II ...
Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
www.elsevier.com/locate/soildyn
Wave propagation in a seven-story reinforced concrete building II. Observed wavenumbers q M.I. Todorovska a,*, S.S. Ivanovic b, M.D. Trifunac a a
Civil Engineering Department, University of Southern California, Los Angeles CA 90089-2531, USA b Civil Engineering Department, University of Montenegro, Podgorica 81000, Yugoslavia Accepted 9 December 2000
Abstract This paper presents estimates of wavenumbers of propagating waves in a seven-story reinforced concrete building in Van Nuys, California, using recorded response to four earthquakes. The phase velocities inferred from these wavenumbers are consistent from one earthquake to another. They are also consistent, inside the building, with independent estimates of the shear wave velocities in the building (e.g. using ambient vibration tests), and along the base, with phase velocities of Love waves typical for San Fernando Valley. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Damage detection; System identi®cation; Wave propagation in structures; Earthquake response analysis
1. Introduction In Part I of this paper [8] we reviewed two-dimensional, ¯exible base, wave propagation models of buildings and presented dispersion analysis of these models for parameters corresponding to an instrumented seven-story reinforced concrete hotel building in Van Nuys, California (VN7SH; [1]). In this paper (Part II), we continue with interpretation of the recorded earthquake response of the same building using wave propagation methods. In particular, we compute approximately horizontal and vertical wavenumbers of propagating waves between two recording points, and look for consistency of the phase velocity inferred from these wavenumbers with other independent estimates. The possibility to estimate phase velocities of propagating waves between two points in the buildings (e.g. adjacent ¯oors) and detect reliably their changes, possibly due to loss of stiffness along the wave path, is potentially valuable for developing new, more powerful, methods for detection of local damage based on wave propagation, rather than on modal methods [2,6]. We use records from four earthq This paper is dedicated to Professor Vlatko BrcÏic (1919±2000), our teacher and mentor, in recognition of his devotion, inspiring teaching, leadership and invaluable contributions to the University of Belgrade. His ideas, goals and quest for the highest standards live on through his grateful students Ð engineers he helped create. * Corresponding author. Tel.: 11-213-740-0616; fax: 11-213-744-1426. E-mail address: [email protected] (M.I. Todorovska).
quakes, including the 1987 Whittier-Narrows (ML 5.9, R 41 km; R epicentral distance), 1992 Landers (ML 7.5, R 186 km), 1992 Big Bear (ML 6.5, R 149 km), and 1994 Northridge (ML 6.4, R 4 km) (Table 1). 2. Earthquake data The ®rst strong motion in the VN7SH building was recorded during the San Fernando earthquake of February 9, 1971 (Fig. 1). We do not use the records because of inadequate location and number of sensors (the instrumentation consisted of three self-contained tri-axial AR-240 accelerographs; [12]). We use records from four other events, listed in Table 1 (for each event, the date, magnitude, epicentral distance and peak horizontal velocity at the ground ¯oor are included). All of these responses were recorded by a 13 channel CR-1 central recording system and one tri-axial SMA-1 accelerograph on the ground ¯oor, both with common trigger mechanism. The sensor locations are shown in Fig. 2. The records of the Whittier-Narrows, Landers, Big Bear and Northridge earthquakes were processed by the California Division of Mines and Geology (CDMG; [7]). 2.1. Wave arrivals from the earthquake sources Fig. 1 shows the location of the VN7SH building in central San Fernando Valley of metropolitan Los Angeles, and the
0267-7261/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0267-726 1(01)00004-5
226
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
Table 1 Selected earthquake and accelerogram parameters describing the data used in this paper Earthquake
Whittier-Narrows Landers Big Bear Northridge a
Date
10/01/87 06/28/92 06/28/92 01/17/94
ML
5.9 7.5 a 6.5 6.4
R (km)
41 186 149 1.5
w (8)
aH,MX (g)
117 90 91.5 270±318 240±350
0.16 0.04 0.01 0.44
aV,MX (g)
± 0.007 0.007 0.27
vmax (cm/s) NS
EW
8.4 11.4 3.9 38.4
6.4 10.6 3.6 50.9
Ms.
Fig. 1. San Fernando Valley area showing position of VN7SH building site relative to the horizontal projections of the fault planes of the 1971 San Fernando and 1994 Northridge earthquakes. The arrows show the directions toward the sources of the 1987 Whittier-Narrows, 1992 Landers and 1992 Big Bear earthquakes.
directions of wave arrivals for all of the four events. The dashed oval surfaces represent the horizontal projections of the San Fernando and Northridge earthquake faults. Fig. 3 shows a pro®le of shear (Vs) and compressional (Vp) wave velocities typical for San Fernando Valley, and Fig. 4 shows phase velocities of the ®rst ®ve Love wave modes computed for this pro®le.
Fig. 2. Locations of channels 1±16 in the VN7SH building.
The San Fernando earthquake faulting began at depth ,9 km below the surface and propagated up along the fault plane, dipping at 408 and striking at N72W, with average dislocation velocity of 2 km/s. A consequence of this rupture propagation is that the earthquake waves were ®rst arriving towards the building site at an angle w 228 (measured clockwise from north), and then at an increasing angle towards w 628 during the following 9 s of faulting (Fig. 1). The strongest motion arrived from the deep part of the dislocation surface, during the ®rst 4 s of faulting, and then from the shallow part of fault during the last 2 s of faulting [9]. During the Northridge earthquake, the largest dislocations propagated during the ®rst 5 s towards northwest, and then during the last 1 s towards west, with dislocation velocity between 2.8 and 3 km/s [18]. Assuming that most of the strong motion energy arrived from the area with the largest dislocation amplitudes, it follows that the waves started to arrive at w 2708, and ended at w 3188. However, because the
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
227
Fig. 4. Phase velocities of the ®rst ®ve Love wave modes in San Fernando Valley. For phase velocities above the dashed line, incident Love waves have horizontal wavelengths longer than four times the building length (4L 160 m).
Fig. 3. Shear Vs and compressional Vp wave velocities in San Fernando Valley.
entire fault surface radiated energy, this range of directions is actually broader, from w 2408 to about w 3508. The epicentral distances to the Whittier-Narrows, Landers and Big Bear earthquakes were 41, 186 and 149 km, respectively. The fault dimensions of these earthquakes were small relative to the respective epicentral distances. In a laterally homogeneous layered earth (Figs. 3 and 4), this would have resulted in all the wave energy arriving along essentially the same respective paths. However, the complex geologic structure, with the Santa Monica and Verdugo Mountains, and Elysian and Repetto Hills between San Gabrial and Los Angeles Valleys, contributed to lateral scattering, resulting in more complicated multiple direction arrivals of the wave energy towards the building. 2.2. Accuracy of digitization To estimate the order of magnitude of the time delay between motions recorded by channels 1 and 13 (both at ground level), we use Fig. 5 which shows a schematic vertical cross section through the focus of the Northridge earthquake and the VN7SH building (see Fig. 1). It is seen that, for waves arriving towards recording stations 1 and 13 (at the western and eastern ends of the building, L 40 m apart), guided SH and Love wave components of the early motions could be delayed by Dt L=c & 40=800 0.05 s. Direct body waves would be delayed less (due to large radial phase velocity CR), while the waves traveling through shallow low velocity layers would be delayed at most by Dt 0.05 s. About 5±6 s later, the energy originates from the northwestern end of the fault, at
Fig. 5. (a) Schematic vertical cross section through the focus of the 1994 Northridge earthquake and the VN7SH building site (see Fig. 1); (b) An illustration how this cross section changed 5±6 s later, towards the end of the main dislocation sequence. Note in (a) the predominantly eastward propagation of energy during 0±1 s, and in (b) the south-eastward propagation 5±6 s later.
about 10 km depth, and about 15 km NW from the VN7SH building. For directions of arrival of direct body waves w 318±3508, `the separation distance' for stations 1 and 13 reduces to ,8 and 33 m, which reduces the time delay between the wave arrivals to these two stations to Dt 0.008 s or shorter. Following the direct arrivals from the fault plane, surface and coda waves of strong motion may arrive from all directions, scattered and re¯ected from the edges of sediments in San Fernando Valley. For the above delays to be seen in the recorded data, it is necessary to have accurate relative timing of the traces recorded by the CR-1 system, which requires accurate digitization of the beginning of the traces. Until the early 1990s, our software for routine processing of strong motion accelerograms produced Volume II data (time histories of
228
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
Fig. 6. Time shifts for the CR-1 records of four earthquakes digitized by CDMG.
acceleration, velocity and displacement corrected for the baseline and for the instrument response; [5]), equally sampled at 50 points per second (Dt 0.02 s). Since 1994, our Volume II data are available at 100 points per second. Consequently, the above described `delays' between motions at stations 1 and 13 are comparable to 1 to several intervals of equally spaced data at Dt 0.01 s (i.e. 0.01±0.03 s). To examine the relative time coordinates of the Northridge earthquake response data for this building, as digitized by CDMG, we redigitized the traces again using as an original a reproduction of these records in Ref. [7]. By visual comparison of the ®rst several seconds of digitized data and by cross-correlation analysis, we found that the CDMG digitization starts late up to ,0.03 s for some traces. In Fig. 6, we show these `skipped' intervals for all the 13 channels. (The channels are ordered from top to bottom as they appear on the ®lm record.) It is seen that the origin times of both versions of the Northridge records agree for channels 1 and 2. For the subsequent channels, the traces digitized by CDMG begin to be progressively late with increasing channel number, and the delay reaches about 0.02 s for channels 8±13. Channel 12 has the largest delay, of about 0.03 s. Fig. 6 also shows similar comparison of initial delays for three other earthquakes: WhittierNarrows, Big Bear and Landers. It is seen that the difference is of systematic and repetitive nature. There are several plausible explanations for these differences. As an illustration only, we suggest that one possible explanation can be related to different optical densities of the traces on ®lm. If all the traces are digitized with the same threshold level of gray, then the trace with smaller optical density would appear as thinner and would `start' a few pixels later [5,15]. For the data used in this paper, whenever we used CDMG digitization, we padded zeroes in the beginning to correct for the above shifts. Without these corrections, many aspects of the following analyses would have been more dif®cult or impossible to conduct.
3. Methodology In the following, we present the theoretical basis for estimation of wavenumbers of propagating waves between two points, and suggest how these can be computed from recorded strong motion in the building. We also discuss the limitations and the accuracy of this procedure. 3.1. Empirical estimation of wavenumbers Motion consisting of a plane SH-wave, with frequency in a narrow band Dv centered at v0 and propagating in the xdirection (i.e. along the longitudinal axis of the building; see Fig. 2), can be represented by v x; t
Zv0 1Dv=2 v0 2Dv=2
v vexpiv t 2 x=c v0 dv
1
where v(v ) is the average of the Fourier transform of v x; t along the building length, L, and c(v ) is the horizontal phase velocity of propagation. Then the derivative of v x; t with respect to x is dv x; t v v x; t i c v0 dx
2
The ratio v=c v0 is the wavenumber kx v0 which can be computed using Eq. (2) as follows kx v0 2i
dv =v dx
3
For long wavelengths l cT q L where T 2p=v, the derivative dv/dx can be approximated by dv x; t Dv < dx D
4
where Dv is the difference between the motion at two recording points and D is the distance between the two
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
points, and v in Eq. (3) can be substituted by the average of the motions at the two points. This can be generalized to computing approximately wavenumbers for vertical and horizontal shear-wave propagation between any pair of stations (i, j). We de®ne vi t 2 vj t vi t 1 vj t 5 Ki;j v ; F =F D 2 where F(´) indicates Fourier Transform. For long wavelengths compared to the separation distance, and for frequency away from the modal frequencies (see Ref. [8]), Ki;j v will be equal approximately to the wavenumber v 6 k v c v of the propagating shear wave between points i and j. In case of the VN7SH building, v will be the displacement in the NS direction. For a propagating wave with constant c, Ki;j v will be a straight line with slope 1/c. For c that is dependent on v , the slope of Ki;j v would be more complicated dKi;j 1 c 0 v 2v 2 dv c v c v
7
but for small v and/or slowly changing c(v ), the second term is small and dKi;j 1 < c v dv
8
Under these conditions, c(v ) can be estimated from the slope of Ki;j v. Changes in the stiffness in the elements between channels i and j will lead to changes in c(v ) and consequently in changes in the slope of Ki;j v. This opens a possibility for development of new methods for detection of local damage based on changes in Ki;j v and its slope. We leave the validation of this thesis against actual data for a future paper. The above procedure for empirical estimation of wavenumbers was derived on the assumption that there are propagating waves in the medium. In the companion paper [8], we presented simple, two-dimensional, ¯exiblebase building models. In the frequency domain, their response was presented as a superposition of the ¯exiblebase modes, some of which were harmonic and others exponential functions. These modes resulted from interference of waves re¯ected from the stress-free boundaries and are standing waves by nature. The phase velocities associated with these modes depended on frequency. The same applied to the wavenumbers. At the ®xed-base frequencies, the ¯exible-base modes became identical to the corresponding ®xed-base modes. Because it takes time for wave interference to occur, in reality the motion in the building consists of standing waves and propagating waves. It is expected that
229
experimentally computed Ki;j v will have the general feature of the propagating wave with superimposed features due to the modes of vibration in the neighborhood of the modal frequencies. 3.2. Limitations and accuracy of the procedure The approximation of wavenumbers by Eq. (5) is valid only for wavelengths l . 4D which implies Ki;j v ,
p 2D
9
For horizontal wave propagation along the ¯oor slabs or at the ground ¯oor, D L 40 m implying Ki;j , 0.04 m 21. For vertical wave propagation between the ®rst and second ¯oors, D h 4.1 m implying Ki;j , 0.4 m 21. Similarly, for vertical wave propagation between the second and third ¯oors, D h 2.7 m implying Ki;j , 0.6 m 21. The accuracy of computing Ki;j by Eq. (5) depends on the accuracy of relative timing of the recorded motions at channels i and j. This is particularly important for ®lm records even though the trigger time is the same for all the channels. In Fig. 7, we illustrate the consequences on computed K1;13 v by shifting v1 t by ^0.02 s. The heaver line shows K1,13 versus frequency f evaluated using u1 t as digitized, and the weaker lines show K1,13 computed using u1 t shifted by ^0.02 s. The bar over the time histories on the top of Fig. 7 shows the length of the segments used to compute K1,13. The weak straight lines show k v v=c computed for constant c 1.0, 1.9, 3.0, 4.3 and 7.0 km/s. It is seen that K1,13 is very sensitive to the relative timing and changes by a factor of 2±3. We conclude that to estimate wavenumbers from analog records of strong motion, it is crucial that the ®rst point of each acceleration trace on the ®lm (corresponding to time t 0) is digitized very accurately. Our automatic digitization system achieves accuracy of one pixel or 0.004 s, for a ®lm image scanned by a ¯at bed scanner with 600 points per inch (236 points/cm) resolution, and using a special algorithm we have developed [15]. 4. Results and discussion We computed Kij using pairs of channels located on the ®rst (ground), second and third levels of the building. We chose this part of the building because no severe structural damage was observed there following the Northridge earthquake [14], and because the purpose of this paper is to ®nd out whether there is a qualitative agreement between phase velocities of propagating waves inferred from wavenumbers and velocities estimated previously (see Ref. [8]) from ambient vibration tests and from material properties. We will present and discuss Kij for the upper (damaged) levels in a future paper, within the context of damage detection using observed Kij .
230
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
x Fig. 7. Plots of K1;13 calculated using the strong motion part of the signals at channels 1 and 13 (ground ¯oor) recorded during the 1994 Northridge earthquake (the part of the signal used is marked in the plots of u on the top). The different curves show the effects of ^0.02 s time delays between the motions at x channels 1 and 13 on K1;13 .
4.1. Observed wavenumbers at ground level Figs. 8±11 show K1,13 computed by Eq. (5) for the Whittier-Narrows, Landers, Big Bear and Northridge earthquakes. On the top, the recorded time histories are shown, with the bars indicating the segment of the record used to compute K1,13. These segments were chosen so that the expected phase velocities are not very large and can be detected from the phase difference between the motions at the two channels. The gray band marks the limit up to which K1,13 is related to the wavenumber kx, described by Eq. (9), and equal to 0.04 m 21 for K1,13. The wavenumbers estimated from channels 1 and 13 would be those of wave propagation in the sediments at the building site if there was no soil±structure interaction and if the building was supported by a ¯exible surface foundation. Because of the soil±structure interaction and the presence of piles, the computed wavenumbers are `contaminated' but should be of same order of magnitude as the ones in the soil [13]. During strong motion caused by direct arrivals of shear waves, the radial phase velocity CR in Fig. 5 can be large,
due to nearly vertical incidence of these waves. This would be the case for the segments 3±12 s for the Northridge and 6±8 s for the Whittier-Narrows earthquakes (time zero is the trigger time). During the subsequent arrival of surface waves and various scattered `surface' waves, the phase velocities should be in the range 0.5±1.5 km/s (see Figs. 3 and 4). This would be the case for the segments 8±20 s for the Whittier-Narrows earthquake, essentially the entire record length for the Landers and Big Bear earthquakes, and ,12± 60 s for the Northridge earthquake. The Northridge earthquake started almost beneath the building and thus had more high frequency energy and `stronger' participation of the higher order surface wave modes. Indeed, we ®nd somewhat larger phase velocities in Fig. 11 for f , 10 Hz [2]. For f typically smaller than 5 Hz, K1,13 for the WhittierNarrows, Landers and Big Bear earthquakes (Figs. 8±10) implies motions associated with `slow' surface waves. Those may be associated with high frequency fundamental Love wave modes, created along the edges of southeastern and eastern San Fernando Valley following these earthquakes, and then propagating westward towards the site. For the Northridge earthquake (Fig. 11), K1,13 implies higher
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
231
x Fig. 8. Plot of K1;13 calculated using the early `coda' of ground motion (8±20 s) recorded during the 1987 Whittier-Narrows earthquake (the part of the signal used is marked in the plots of u on the top).
phase velocities, suggesting stronger participation of the higher Love wave modes and of direct body waves. The results in Figs. 8±11 imply apparent EW velocities in the range 500±2500 m/s, which is in qualitative agreement with Love wave phase velocities computed for the soil and sediments underlying the building (Fig. 4). In our previous work [2], we showed that the observed delays between the recording channels 1 and 13 are also consistent with the expected direction of wave approach during the same earthquakes. Finally, the low amplitude and ambient vibration tests of VN7SH suggested horizontal and vertical warping of the foundation, accompanying the wave propagation of ground noise, which is believed to consist primarily of high frequency Rayleigh waves, and results in apparent surface phase velocities of about 300 m/s [13]. 4.2. Observed wavenumbers in the building In Figs. 12±15, we show Ki;j computed for vertical and horizontal wave propagation between seven combinations of channels, all located at levels one, two or three. The plots on the left and right sides show, respectively, Ki;j for vertical wave propagation along the west and east columns of the building, between the ®rst and second ¯oors
(bottom), and between the second and third ¯oors (top). The straight lines correspond to wavenumbers k v=c corresponding to constant c cz 50 and 100 m/s. The horizontal shaded lines correspond to Ki;j 0.4 and 0.6 m 21, respectively, for the plots on the bottom and top, which follows from the constraint Ki;j , p/2D, where D is the interstory height (see Section 3.2). It can be seen that Ki;j has peaks (related to standing waves, i.e. the modes of vibration) and a backbone (related to direct wave propagation) with slope increasing with frequency, roughly linearly, and indicating vertical phase velocities cz , 50 2 200 m/s for all four earthquakes. These values are in qualitative agreement with the estimates of the vertical shear wave velocity b z based on the collective stiffness of the columns and shear walls (122 m/s; Eqs. (28) and (29) in Ref. [8]), and based on the observed apparent system periods during ambient vibration tests (112 m/s for NS vibrations; Eq. (19) in Ref. [8]). Because the latter is an estimate based on the period of the soil±structure system, rather than on the ®xed-base period of the building, it is a lower bound of b z. The plots in the middle column show Ki;j for horizontal wave propagation, at ground level (bottom), and along the slabs of the second (middle) and third (top) ¯oors. The straight lines show k v /c for constant c cz 500,
232
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
x Fig. 9. Plot of K1;13 calculated using the ®rst 43 s of ground motion during the 1992 Landers earthquake.
1000 and 2000 m/s. The slope of the backbone of Ki;j in these plots indicates that, along the ¯oor slabs, cx , 1000± 2000 m/s, which is again consistent with simpli®ed theoretical estimates (2049 m/s; Eq. (30) in Ref. [8]) and approximately agrees for all four earthquakes. The peaks in Ki;j correspond to standing waves associated with the apparent system frequencies of the building±foundation±soil system. The ambient vibration tests gave f 1.4 and 1.6 Hz for the ®rst NS translational and ®rst torsional modes, and f 4.2 and 4.9 Hz for the second NS translational and second torsional mode [3,4]. In Refs. [16,17], we showed that the frequency of the ®rst translational mode decreased with increasing level of shaking. Indeed, we notice in the plots of the vertical Ki;j a shift of the ®rst peak towards lower f for the Northridge earthquake which shook the building the most. The other system frequencies do not appear regularly (at the same frequency) for different earthquake excitations. A more detailed analysis of these peaks is beyond the scope of this paper. 5. Summary and conclusions We presented plots of observed wavenumbers Ki;j v for seven pairs of channels (i, j) in the VN7SH building, located
up to and including the third ¯oor, during four earthquakes. This part of the building was not damaged by any of these events. The observed Ki;j v was computed using Eq. (5), which is based on the assumption that there is a propagating wave between the two recording points, and is valid for wavenumbers k , p/2D, where D is the separation distance. This implies k , 0.04 m 21 for horizontal wave propagation across the length of the building, k , 0.4 m 21 for vertical wave propagation between the ®rst and second ¯oors, and k , 0.6 m 21 for vertical wave propagation between all other adjacent ¯oors. Because re¯ection of an incident wave from the boundaries of the building is not instantaneous (it takes time for the waves to reach the boundaries), the motion in the building consists of both standing and propagating waves (the standing waves are the modes of vibration). The recorded motions contain both of these two types of waves. Consequently, the computed Ki;j v has peaks associated with the modes of vibration (near the modal frequencies), but its backbone is roughly a straight line, with slope characterizing the propagating waves. The slope of Ki;j v is ,v /c, and can be used to estimate the phase velocity c of the propagating wave. The purpose of this study was to ®nd out whether there is at least a qualitative agreement between the phase velocities of the propagating waves in the columns and slabs,
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
233
x Fig. 10. Plot of K1;13 calculated using the ®rst 20 s of ground motion during the 1992 Big Bear earthquake.
estimated from the backbone of Ki;j , from one event to another, and with independent estimates, e.g. from the frequency of the ®rst NS mode of vibration measured by ambient vibration tests, or from the material properties and design characteristics. This is potentially valuable for structural health monitoring and damage detection using wave propagation methods, because local changes in c, inferred from observed Ki;j v between two ¯oors, for example, may indicate local changes in stiffness due to damage. This method would be superior to the modal methods, because the measured modal frequencies (and their changes) depend on the overall stiffness of the soil±structure system, while c (and its changes) can be measured locally using wave propagation methods and is not affected by soil±structure interaction. However, this would require dense instrumentation of the building [10,11], recording at a high sampling rate, and accurate timing. The following are our conclusions and recommendations based on the presented results. 1. The inferred c for vertical wave propagation along the end columns is consistent from one earthquake to another and agrees qualitatively with the vertical shear wave velocity for NS vibrations estimated from ambient vibra-
tion tests (b z 112 m/s) and from the collective stiffness of the columns (b z 122 m/s). 2. The inferred c for horizontal wave propagation along the ¯oor slabs is consistent from one earthquake to another and agrees qualitatively with the horizontal shear wave velocity through the slabs estimated from the material properties of concrete (b z 2049 m/s). 3. At the ground ¯oor, along the longitudinal axis of the building, the inferred c for horizontal wave propagation between the two ends of the building (see Fig. 2) indicates phase velocities in the range 0.5±2.5 km/s. This is in qualitative agreement with Love wave phase velocities computed for the soil and sediments underlying the building using a parallel layer model (Fig. 4). 4. For analog records, the accuracy of computing Ki;j depends critically on the accuracy with which the records at channels i and j are synchronized. For analog records (most mainshock building records available at this time are analog), this translates to the accuracy of picking up the ®rst digitized point on the trace (corresponding to trigger time), which is not obvious because of different optical density of each trace. Using our special algorithm (part of our LeAuto accelerogram digitization package) this accuracy is one
234
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
x Fig. 11. Plot of K1;13 calculated using the early `coda' part of the signal (12±60 s) recorded during the 1994 Northridge earthquake.
z x Fig. 12. Plots of Ki;j and Ki;j computed for the 1987 Whittier-Narrows earthquake from six pairs of channels located at ¯oors 1 (ground) to 3.
pixel, i.e. 1/236 s for a 600 dots per inch ¯at-bed scanner and 1 cm/s ®lm speed [15]. 5. Local changes in c, estimated as in this paper, may indicate local loss of stiffness due to damage. A similar
analysis for the upper ¯oors of this building, where structural damage was observed following the Northridge earthquake may con®rm the feasibility of this method for damage detection. We will present this in a future paper.
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
Fig. 13. Same as Fig. 12 but for the 1992 Landers earthquake.
Fig. 14. Same as Fig. 12 but for the 1992 Big Bear earthquake.
Fig. 15. Same as Fig. 12 but for the 1994 Northridge earthquake.
235
236
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
References [1] Blume JA et al. In: San Fernando, California Earthquake of February 9, 1971, vol. 1, Part A, p. 359±393. Washington, DC: US Department of Commerce, National Oceanic and Atmospheric Administration, 1973. [2] Ivanovic SS, Trifunac MD, Todorovska MI. On identi®cation of damage in structures via wave travel times. In: Proc NATO Advanced Research Workshop on Strong Motion Instrumentation for Civil Engineering Structures, Istanbul, 2±5 June, 1999. [3] Ivanovic SS, Novikova EI, Gladkov AA, Todorovska MI. Ambient vibration tests of a seven story reinforced concrete building damaged by the 1994 Northridge, California earthquake. Soil Dynam Earthquake Engng 2000;19:391±411. [4] Ivanovic SS, Trifunac MD, Todorovska MI. Ambient vibration tests of structuresÐa review. Indian Society of Earthquake Technology Journal 2000;37: in press. [5] Lee VW, Trifunac MD. Automatic digitization and processing of accelerograms using PC. Department of Civil Engineering, Report No. 90-03, University of Southern California, Los Angeles, California, 1990. [6] Safak E. Detection of seismic damage in multi-story buildings by using wave propagation analysis. Proc Sixth US National Conf on Earthquake Engineering, Seattle, Washington, 1998. [7] Shakal A, Huang M, Darragh R, Cao T, Sherburne R, Malhotra P, Cramer C, Syndov R, Graizer V, Maldonado G, Peterson C, Wimpole J. CSMIP strong motions records from the Northridge, California, earthquake of 17 January 1994. Report no. OSMS 94-07, California Dept of Conservation, Div of Miner and Geology, Sacramento, California. [8] Todorovska MI, Ivanovic SS, Trifunac MD. Wave propagation in a seven-story reinforced concrete building Part I: theoretical models. Soil Dynam Earthquake Engng 2001;21:211±23. [9] Trifunac MD. A three dimensional dislocation model for the San
[10]
[11]
[12] [13]
[14]
[15] [16] [17] [18]
Fernando, California, earthquake of February 9, 1971. Bull Seism Soc Amer 1974;64:149±72. Trifunac MD, Todorovska MI. Recording and interpreting earthquake response of full-scale structures. Proc NATO Workshop on Strong Motion Instrumentation for Civil Engineering Structures, Istanbul, June 2±5, 1999. Kluwer 1999. Trifunac MD, Todorovska MI. Evolution of accelerographs, data processing, strong motion arrays and amplitude and spatial resolution in recording strong earthquake motion. Soil Dynam Earthquake Engng (submitted for publication). Trifunac MD, Brady AG, Hudson DE. Strong motion earthquake accelerograms, digitized and plotted data, vol. II, Part C, EERL-7251. Pasadena, California: California Institute of Tech, 1973. Trifunac MD, Ivanovic SS, Todorovska MI, Novikova EI, Gladkov AA. Experimental evidence for ¯exibility of a building foundation supported by concrete friction piles. Soil Dynam Earthquake Engng 1999;18:169±87. Trifunac MD, Ivanovic SS, Todorovska MI. Seven story reinforced concrete building in Van Nuys, California: strong motion data recorded between 7 February 1971 and 9 December 1994, and description of damage following the Northridge 17 January 1994 earthquake. Report no. CE 99-02, Dept. of Civil Eng., University of Southern California, Los Angeles, California, 1999. Trifunac MD, Lee VW, Todorovska MI. Common problems in automatic digitization of strong motion accelerograms. Soil Dynam Earthquake Engng 1999;18(7):519±30. Trifunac MD, Ivanovic SS, Todorovska MI. Apparent periods of a building, part I: Fourier analysis. J Struct Engng ASCE 2001;May: in press. Trifunac MD, Ivanovic SS, Todorovska MI. Apparent periods of a building, part II: time-frequency analysis. J Struct Engng ASCE 2001;May: in press. Wald DT, Heaton TH. The slip history of the 1994 Northridge, California, earthquake determined from strong-motion, teleseismic, GPS and leveling data. Bull Seism Soc Amer 1996;86(1B):549±70.
www.elsevier.com/locate/soildyn
Wave propagation in a seven-story reinforced concrete building II. Observed wavenumbers q M.I. Todorovska a,*, S.S. Ivanovic b, M.D. Trifunac a a
Civil Engineering Department, University of Southern California, Los Angeles CA 90089-2531, USA b Civil Engineering Department, University of Montenegro, Podgorica 81000, Yugoslavia Accepted 9 December 2000
Abstract This paper presents estimates of wavenumbers of propagating waves in a seven-story reinforced concrete building in Van Nuys, California, using recorded response to four earthquakes. The phase velocities inferred from these wavenumbers are consistent from one earthquake to another. They are also consistent, inside the building, with independent estimates of the shear wave velocities in the building (e.g. using ambient vibration tests), and along the base, with phase velocities of Love waves typical for San Fernando Valley. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Damage detection; System identi®cation; Wave propagation in structures; Earthquake response analysis
1. Introduction In Part I of this paper [8] we reviewed two-dimensional, ¯exible base, wave propagation models of buildings and presented dispersion analysis of these models for parameters corresponding to an instrumented seven-story reinforced concrete hotel building in Van Nuys, California (VN7SH; [1]). In this paper (Part II), we continue with interpretation of the recorded earthquake response of the same building using wave propagation methods. In particular, we compute approximately horizontal and vertical wavenumbers of propagating waves between two recording points, and look for consistency of the phase velocity inferred from these wavenumbers with other independent estimates. The possibility to estimate phase velocities of propagating waves between two points in the buildings (e.g. adjacent ¯oors) and detect reliably their changes, possibly due to loss of stiffness along the wave path, is potentially valuable for developing new, more powerful, methods for detection of local damage based on wave propagation, rather than on modal methods [2,6]. We use records from four earthq This paper is dedicated to Professor Vlatko BrcÏic (1919±2000), our teacher and mentor, in recognition of his devotion, inspiring teaching, leadership and invaluable contributions to the University of Belgrade. His ideas, goals and quest for the highest standards live on through his grateful students Ð engineers he helped create. * Corresponding author. Tel.: 11-213-740-0616; fax: 11-213-744-1426. E-mail address: [email protected] (M.I. Todorovska).
quakes, including the 1987 Whittier-Narrows (ML 5.9, R 41 km; R epicentral distance), 1992 Landers (ML 7.5, R 186 km), 1992 Big Bear (ML 6.5, R 149 km), and 1994 Northridge (ML 6.4, R 4 km) (Table 1). 2. Earthquake data The ®rst strong motion in the VN7SH building was recorded during the San Fernando earthquake of February 9, 1971 (Fig. 1). We do not use the records because of inadequate location and number of sensors (the instrumentation consisted of three self-contained tri-axial AR-240 accelerographs; [12]). We use records from four other events, listed in Table 1 (for each event, the date, magnitude, epicentral distance and peak horizontal velocity at the ground ¯oor are included). All of these responses were recorded by a 13 channel CR-1 central recording system and one tri-axial SMA-1 accelerograph on the ground ¯oor, both with common trigger mechanism. The sensor locations are shown in Fig. 2. The records of the Whittier-Narrows, Landers, Big Bear and Northridge earthquakes were processed by the California Division of Mines and Geology (CDMG; [7]). 2.1. Wave arrivals from the earthquake sources Fig. 1 shows the location of the VN7SH building in central San Fernando Valley of metropolitan Los Angeles, and the
0267-7261/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0267-726 1(01)00004-5
226
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
Table 1 Selected earthquake and accelerogram parameters describing the data used in this paper Earthquake
Whittier-Narrows Landers Big Bear Northridge a
Date
10/01/87 06/28/92 06/28/92 01/17/94
ML
5.9 7.5 a 6.5 6.4
R (km)
41 186 149 1.5
w (8)
aH,MX (g)
117 90 91.5 270±318 240±350
0.16 0.04 0.01 0.44
aV,MX (g)
± 0.007 0.007 0.27
vmax (cm/s) NS
EW
8.4 11.4 3.9 38.4
6.4 10.6 3.6 50.9
Ms.
Fig. 1. San Fernando Valley area showing position of VN7SH building site relative to the horizontal projections of the fault planes of the 1971 San Fernando and 1994 Northridge earthquakes. The arrows show the directions toward the sources of the 1987 Whittier-Narrows, 1992 Landers and 1992 Big Bear earthquakes.
directions of wave arrivals for all of the four events. The dashed oval surfaces represent the horizontal projections of the San Fernando and Northridge earthquake faults. Fig. 3 shows a pro®le of shear (Vs) and compressional (Vp) wave velocities typical for San Fernando Valley, and Fig. 4 shows phase velocities of the ®rst ®ve Love wave modes computed for this pro®le.
Fig. 2. Locations of channels 1±16 in the VN7SH building.
The San Fernando earthquake faulting began at depth ,9 km below the surface and propagated up along the fault plane, dipping at 408 and striking at N72W, with average dislocation velocity of 2 km/s. A consequence of this rupture propagation is that the earthquake waves were ®rst arriving towards the building site at an angle w 228 (measured clockwise from north), and then at an increasing angle towards w 628 during the following 9 s of faulting (Fig. 1). The strongest motion arrived from the deep part of the dislocation surface, during the ®rst 4 s of faulting, and then from the shallow part of fault during the last 2 s of faulting [9]. During the Northridge earthquake, the largest dislocations propagated during the ®rst 5 s towards northwest, and then during the last 1 s towards west, with dislocation velocity between 2.8 and 3 km/s [18]. Assuming that most of the strong motion energy arrived from the area with the largest dislocation amplitudes, it follows that the waves started to arrive at w 2708, and ended at w 3188. However, because the
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
227
Fig. 4. Phase velocities of the ®rst ®ve Love wave modes in San Fernando Valley. For phase velocities above the dashed line, incident Love waves have horizontal wavelengths longer than four times the building length (4L 160 m).
Fig. 3. Shear Vs and compressional Vp wave velocities in San Fernando Valley.
entire fault surface radiated energy, this range of directions is actually broader, from w 2408 to about w 3508. The epicentral distances to the Whittier-Narrows, Landers and Big Bear earthquakes were 41, 186 and 149 km, respectively. The fault dimensions of these earthquakes were small relative to the respective epicentral distances. In a laterally homogeneous layered earth (Figs. 3 and 4), this would have resulted in all the wave energy arriving along essentially the same respective paths. However, the complex geologic structure, with the Santa Monica and Verdugo Mountains, and Elysian and Repetto Hills between San Gabrial and Los Angeles Valleys, contributed to lateral scattering, resulting in more complicated multiple direction arrivals of the wave energy towards the building. 2.2. Accuracy of digitization To estimate the order of magnitude of the time delay between motions recorded by channels 1 and 13 (both at ground level), we use Fig. 5 which shows a schematic vertical cross section through the focus of the Northridge earthquake and the VN7SH building (see Fig. 1). It is seen that, for waves arriving towards recording stations 1 and 13 (at the western and eastern ends of the building, L 40 m apart), guided SH and Love wave components of the early motions could be delayed by Dt L=c & 40=800 0.05 s. Direct body waves would be delayed less (due to large radial phase velocity CR), while the waves traveling through shallow low velocity layers would be delayed at most by Dt 0.05 s. About 5±6 s later, the energy originates from the northwestern end of the fault, at
Fig. 5. (a) Schematic vertical cross section through the focus of the 1994 Northridge earthquake and the VN7SH building site (see Fig. 1); (b) An illustration how this cross section changed 5±6 s later, towards the end of the main dislocation sequence. Note in (a) the predominantly eastward propagation of energy during 0±1 s, and in (b) the south-eastward propagation 5±6 s later.
about 10 km depth, and about 15 km NW from the VN7SH building. For directions of arrival of direct body waves w 318±3508, `the separation distance' for stations 1 and 13 reduces to ,8 and 33 m, which reduces the time delay between the wave arrivals to these two stations to Dt 0.008 s or shorter. Following the direct arrivals from the fault plane, surface and coda waves of strong motion may arrive from all directions, scattered and re¯ected from the edges of sediments in San Fernando Valley. For the above delays to be seen in the recorded data, it is necessary to have accurate relative timing of the traces recorded by the CR-1 system, which requires accurate digitization of the beginning of the traces. Until the early 1990s, our software for routine processing of strong motion accelerograms produced Volume II data (time histories of
228
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
Fig. 6. Time shifts for the CR-1 records of four earthquakes digitized by CDMG.
acceleration, velocity and displacement corrected for the baseline and for the instrument response; [5]), equally sampled at 50 points per second (Dt 0.02 s). Since 1994, our Volume II data are available at 100 points per second. Consequently, the above described `delays' between motions at stations 1 and 13 are comparable to 1 to several intervals of equally spaced data at Dt 0.01 s (i.e. 0.01±0.03 s). To examine the relative time coordinates of the Northridge earthquake response data for this building, as digitized by CDMG, we redigitized the traces again using as an original a reproduction of these records in Ref. [7]. By visual comparison of the ®rst several seconds of digitized data and by cross-correlation analysis, we found that the CDMG digitization starts late up to ,0.03 s for some traces. In Fig. 6, we show these `skipped' intervals for all the 13 channels. (The channels are ordered from top to bottom as they appear on the ®lm record.) It is seen that the origin times of both versions of the Northridge records agree for channels 1 and 2. For the subsequent channels, the traces digitized by CDMG begin to be progressively late with increasing channel number, and the delay reaches about 0.02 s for channels 8±13. Channel 12 has the largest delay, of about 0.03 s. Fig. 6 also shows similar comparison of initial delays for three other earthquakes: WhittierNarrows, Big Bear and Landers. It is seen that the difference is of systematic and repetitive nature. There are several plausible explanations for these differences. As an illustration only, we suggest that one possible explanation can be related to different optical densities of the traces on ®lm. If all the traces are digitized with the same threshold level of gray, then the trace with smaller optical density would appear as thinner and would `start' a few pixels later [5,15]. For the data used in this paper, whenever we used CDMG digitization, we padded zeroes in the beginning to correct for the above shifts. Without these corrections, many aspects of the following analyses would have been more dif®cult or impossible to conduct.
3. Methodology In the following, we present the theoretical basis for estimation of wavenumbers of propagating waves between two points, and suggest how these can be computed from recorded strong motion in the building. We also discuss the limitations and the accuracy of this procedure. 3.1. Empirical estimation of wavenumbers Motion consisting of a plane SH-wave, with frequency in a narrow band Dv centered at v0 and propagating in the xdirection (i.e. along the longitudinal axis of the building; see Fig. 2), can be represented by v x; t
Zv0 1Dv=2 v0 2Dv=2
v vexpiv t 2 x=c v0 dv
1
where v(v ) is the average of the Fourier transform of v x; t along the building length, L, and c(v ) is the horizontal phase velocity of propagation. Then the derivative of v x; t with respect to x is dv x; t v v x; t i c v0 dx
2
The ratio v=c v0 is the wavenumber kx v0 which can be computed using Eq. (2) as follows kx v0 2i
dv =v dx
3
For long wavelengths l cT q L where T 2p=v, the derivative dv/dx can be approximated by dv x; t Dv < dx D
4
where Dv is the difference between the motion at two recording points and D is the distance between the two
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
points, and v in Eq. (3) can be substituted by the average of the motions at the two points. This can be generalized to computing approximately wavenumbers for vertical and horizontal shear-wave propagation between any pair of stations (i, j). We de®ne vi t 2 vj t vi t 1 vj t 5 Ki;j v ; F =F D 2 where F(´) indicates Fourier Transform. For long wavelengths compared to the separation distance, and for frequency away from the modal frequencies (see Ref. [8]), Ki;j v will be equal approximately to the wavenumber v 6 k v c v of the propagating shear wave between points i and j. In case of the VN7SH building, v will be the displacement in the NS direction. For a propagating wave with constant c, Ki;j v will be a straight line with slope 1/c. For c that is dependent on v , the slope of Ki;j v would be more complicated dKi;j 1 c 0 v 2v 2 dv c v c v
7
but for small v and/or slowly changing c(v ), the second term is small and dKi;j 1 < c v dv
8
Under these conditions, c(v ) can be estimated from the slope of Ki;j v. Changes in the stiffness in the elements between channels i and j will lead to changes in c(v ) and consequently in changes in the slope of Ki;j v. This opens a possibility for development of new methods for detection of local damage based on changes in Ki;j v and its slope. We leave the validation of this thesis against actual data for a future paper. The above procedure for empirical estimation of wavenumbers was derived on the assumption that there are propagating waves in the medium. In the companion paper [8], we presented simple, two-dimensional, ¯exiblebase building models. In the frequency domain, their response was presented as a superposition of the ¯exiblebase modes, some of which were harmonic and others exponential functions. These modes resulted from interference of waves re¯ected from the stress-free boundaries and are standing waves by nature. The phase velocities associated with these modes depended on frequency. The same applied to the wavenumbers. At the ®xed-base frequencies, the ¯exible-base modes became identical to the corresponding ®xed-base modes. Because it takes time for wave interference to occur, in reality the motion in the building consists of standing waves and propagating waves. It is expected that
229
experimentally computed Ki;j v will have the general feature of the propagating wave with superimposed features due to the modes of vibration in the neighborhood of the modal frequencies. 3.2. Limitations and accuracy of the procedure The approximation of wavenumbers by Eq. (5) is valid only for wavelengths l . 4D which implies Ki;j v ,
p 2D
9
For horizontal wave propagation along the ¯oor slabs or at the ground ¯oor, D L 40 m implying Ki;j , 0.04 m 21. For vertical wave propagation between the ®rst and second ¯oors, D h 4.1 m implying Ki;j , 0.4 m 21. Similarly, for vertical wave propagation between the second and third ¯oors, D h 2.7 m implying Ki;j , 0.6 m 21. The accuracy of computing Ki;j by Eq. (5) depends on the accuracy of relative timing of the recorded motions at channels i and j. This is particularly important for ®lm records even though the trigger time is the same for all the channels. In Fig. 7, we illustrate the consequences on computed K1;13 v by shifting v1 t by ^0.02 s. The heaver line shows K1,13 versus frequency f evaluated using u1 t as digitized, and the weaker lines show K1,13 computed using u1 t shifted by ^0.02 s. The bar over the time histories on the top of Fig. 7 shows the length of the segments used to compute K1,13. The weak straight lines show k v v=c computed for constant c 1.0, 1.9, 3.0, 4.3 and 7.0 km/s. It is seen that K1,13 is very sensitive to the relative timing and changes by a factor of 2±3. We conclude that to estimate wavenumbers from analog records of strong motion, it is crucial that the ®rst point of each acceleration trace on the ®lm (corresponding to time t 0) is digitized very accurately. Our automatic digitization system achieves accuracy of one pixel or 0.004 s, for a ®lm image scanned by a ¯at bed scanner with 600 points per inch (236 points/cm) resolution, and using a special algorithm we have developed [15]. 4. Results and discussion We computed Kij using pairs of channels located on the ®rst (ground), second and third levels of the building. We chose this part of the building because no severe structural damage was observed there following the Northridge earthquake [14], and because the purpose of this paper is to ®nd out whether there is a qualitative agreement between phase velocities of propagating waves inferred from wavenumbers and velocities estimated previously (see Ref. [8]) from ambient vibration tests and from material properties. We will present and discuss Kij for the upper (damaged) levels in a future paper, within the context of damage detection using observed Kij .
230
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
x Fig. 7. Plots of K1;13 calculated using the strong motion part of the signals at channels 1 and 13 (ground ¯oor) recorded during the 1994 Northridge earthquake (the part of the signal used is marked in the plots of u on the top). The different curves show the effects of ^0.02 s time delays between the motions at x channels 1 and 13 on K1;13 .
4.1. Observed wavenumbers at ground level Figs. 8±11 show K1,13 computed by Eq. (5) for the Whittier-Narrows, Landers, Big Bear and Northridge earthquakes. On the top, the recorded time histories are shown, with the bars indicating the segment of the record used to compute K1,13. These segments were chosen so that the expected phase velocities are not very large and can be detected from the phase difference between the motions at the two channels. The gray band marks the limit up to which K1,13 is related to the wavenumber kx, described by Eq. (9), and equal to 0.04 m 21 for K1,13. The wavenumbers estimated from channels 1 and 13 would be those of wave propagation in the sediments at the building site if there was no soil±structure interaction and if the building was supported by a ¯exible surface foundation. Because of the soil±structure interaction and the presence of piles, the computed wavenumbers are `contaminated' but should be of same order of magnitude as the ones in the soil [13]. During strong motion caused by direct arrivals of shear waves, the radial phase velocity CR in Fig. 5 can be large,
due to nearly vertical incidence of these waves. This would be the case for the segments 3±12 s for the Northridge and 6±8 s for the Whittier-Narrows earthquakes (time zero is the trigger time). During the subsequent arrival of surface waves and various scattered `surface' waves, the phase velocities should be in the range 0.5±1.5 km/s (see Figs. 3 and 4). This would be the case for the segments 8±20 s for the Whittier-Narrows earthquake, essentially the entire record length for the Landers and Big Bear earthquakes, and ,12± 60 s for the Northridge earthquake. The Northridge earthquake started almost beneath the building and thus had more high frequency energy and `stronger' participation of the higher order surface wave modes. Indeed, we ®nd somewhat larger phase velocities in Fig. 11 for f , 10 Hz [2]. For f typically smaller than 5 Hz, K1,13 for the WhittierNarrows, Landers and Big Bear earthquakes (Figs. 8±10) implies motions associated with `slow' surface waves. Those may be associated with high frequency fundamental Love wave modes, created along the edges of southeastern and eastern San Fernando Valley following these earthquakes, and then propagating westward towards the site. For the Northridge earthquake (Fig. 11), K1,13 implies higher
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
231
x Fig. 8. Plot of K1;13 calculated using the early `coda' of ground motion (8±20 s) recorded during the 1987 Whittier-Narrows earthquake (the part of the signal used is marked in the plots of u on the top).
phase velocities, suggesting stronger participation of the higher Love wave modes and of direct body waves. The results in Figs. 8±11 imply apparent EW velocities in the range 500±2500 m/s, which is in qualitative agreement with Love wave phase velocities computed for the soil and sediments underlying the building (Fig. 4). In our previous work [2], we showed that the observed delays between the recording channels 1 and 13 are also consistent with the expected direction of wave approach during the same earthquakes. Finally, the low amplitude and ambient vibration tests of VN7SH suggested horizontal and vertical warping of the foundation, accompanying the wave propagation of ground noise, which is believed to consist primarily of high frequency Rayleigh waves, and results in apparent surface phase velocities of about 300 m/s [13]. 4.2. Observed wavenumbers in the building In Figs. 12±15, we show Ki;j computed for vertical and horizontal wave propagation between seven combinations of channels, all located at levels one, two or three. The plots on the left and right sides show, respectively, Ki;j for vertical wave propagation along the west and east columns of the building, between the ®rst and second ¯oors
(bottom), and between the second and third ¯oors (top). The straight lines correspond to wavenumbers k v=c corresponding to constant c cz 50 and 100 m/s. The horizontal shaded lines correspond to Ki;j 0.4 and 0.6 m 21, respectively, for the plots on the bottom and top, which follows from the constraint Ki;j , p/2D, where D is the interstory height (see Section 3.2). It can be seen that Ki;j has peaks (related to standing waves, i.e. the modes of vibration) and a backbone (related to direct wave propagation) with slope increasing with frequency, roughly linearly, and indicating vertical phase velocities cz , 50 2 200 m/s for all four earthquakes. These values are in qualitative agreement with the estimates of the vertical shear wave velocity b z based on the collective stiffness of the columns and shear walls (122 m/s; Eqs. (28) and (29) in Ref. [8]), and based on the observed apparent system periods during ambient vibration tests (112 m/s for NS vibrations; Eq. (19) in Ref. [8]). Because the latter is an estimate based on the period of the soil±structure system, rather than on the ®xed-base period of the building, it is a lower bound of b z. The plots in the middle column show Ki;j for horizontal wave propagation, at ground level (bottom), and along the slabs of the second (middle) and third (top) ¯oors. The straight lines show k v /c for constant c cz 500,
232
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
x Fig. 9. Plot of K1;13 calculated using the ®rst 43 s of ground motion during the 1992 Landers earthquake.
1000 and 2000 m/s. The slope of the backbone of Ki;j in these plots indicates that, along the ¯oor slabs, cx , 1000± 2000 m/s, which is again consistent with simpli®ed theoretical estimates (2049 m/s; Eq. (30) in Ref. [8]) and approximately agrees for all four earthquakes. The peaks in Ki;j correspond to standing waves associated with the apparent system frequencies of the building±foundation±soil system. The ambient vibration tests gave f 1.4 and 1.6 Hz for the ®rst NS translational and ®rst torsional modes, and f 4.2 and 4.9 Hz for the second NS translational and second torsional mode [3,4]. In Refs. [16,17], we showed that the frequency of the ®rst translational mode decreased with increasing level of shaking. Indeed, we notice in the plots of the vertical Ki;j a shift of the ®rst peak towards lower f for the Northridge earthquake which shook the building the most. The other system frequencies do not appear regularly (at the same frequency) for different earthquake excitations. A more detailed analysis of these peaks is beyond the scope of this paper. 5. Summary and conclusions We presented plots of observed wavenumbers Ki;j v for seven pairs of channels (i, j) in the VN7SH building, located
up to and including the third ¯oor, during four earthquakes. This part of the building was not damaged by any of these events. The observed Ki;j v was computed using Eq. (5), which is based on the assumption that there is a propagating wave between the two recording points, and is valid for wavenumbers k , p/2D, where D is the separation distance. This implies k , 0.04 m 21 for horizontal wave propagation across the length of the building, k , 0.4 m 21 for vertical wave propagation between the ®rst and second ¯oors, and k , 0.6 m 21 for vertical wave propagation between all other adjacent ¯oors. Because re¯ection of an incident wave from the boundaries of the building is not instantaneous (it takes time for the waves to reach the boundaries), the motion in the building consists of both standing and propagating waves (the standing waves are the modes of vibration). The recorded motions contain both of these two types of waves. Consequently, the computed Ki;j v has peaks associated with the modes of vibration (near the modal frequencies), but its backbone is roughly a straight line, with slope characterizing the propagating waves. The slope of Ki;j v is ,v /c, and can be used to estimate the phase velocity c of the propagating wave. The purpose of this study was to ®nd out whether there is at least a qualitative agreement between the phase velocities of the propagating waves in the columns and slabs,
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
233
x Fig. 10. Plot of K1;13 calculated using the ®rst 20 s of ground motion during the 1992 Big Bear earthquake.
estimated from the backbone of Ki;j , from one event to another, and with independent estimates, e.g. from the frequency of the ®rst NS mode of vibration measured by ambient vibration tests, or from the material properties and design characteristics. This is potentially valuable for structural health monitoring and damage detection using wave propagation methods, because local changes in c, inferred from observed Ki;j v between two ¯oors, for example, may indicate local changes in stiffness due to damage. This method would be superior to the modal methods, because the measured modal frequencies (and their changes) depend on the overall stiffness of the soil±structure system, while c (and its changes) can be measured locally using wave propagation methods and is not affected by soil±structure interaction. However, this would require dense instrumentation of the building [10,11], recording at a high sampling rate, and accurate timing. The following are our conclusions and recommendations based on the presented results. 1. The inferred c for vertical wave propagation along the end columns is consistent from one earthquake to another and agrees qualitatively with the vertical shear wave velocity for NS vibrations estimated from ambient vibra-
tion tests (b z 112 m/s) and from the collective stiffness of the columns (b z 122 m/s). 2. The inferred c for horizontal wave propagation along the ¯oor slabs is consistent from one earthquake to another and agrees qualitatively with the horizontal shear wave velocity through the slabs estimated from the material properties of concrete (b z 2049 m/s). 3. At the ground ¯oor, along the longitudinal axis of the building, the inferred c for horizontal wave propagation between the two ends of the building (see Fig. 2) indicates phase velocities in the range 0.5±2.5 km/s. This is in qualitative agreement with Love wave phase velocities computed for the soil and sediments underlying the building using a parallel layer model (Fig. 4). 4. For analog records, the accuracy of computing Ki;j depends critically on the accuracy with which the records at channels i and j are synchronized. For analog records (most mainshock building records available at this time are analog), this translates to the accuracy of picking up the ®rst digitized point on the trace (corresponding to trigger time), which is not obvious because of different optical density of each trace. Using our special algorithm (part of our LeAuto accelerogram digitization package) this accuracy is one
234
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
x Fig. 11. Plot of K1;13 calculated using the early `coda' part of the signal (12±60 s) recorded during the 1994 Northridge earthquake.
z x Fig. 12. Plots of Ki;j and Ki;j computed for the 1987 Whittier-Narrows earthquake from six pairs of channels located at ¯oors 1 (ground) to 3.
pixel, i.e. 1/236 s for a 600 dots per inch ¯at-bed scanner and 1 cm/s ®lm speed [15]. 5. Local changes in c, estimated as in this paper, may indicate local loss of stiffness due to damage. A similar
analysis for the upper ¯oors of this building, where structural damage was observed following the Northridge earthquake may con®rm the feasibility of this method for damage detection. We will present this in a future paper.
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
Fig. 13. Same as Fig. 12 but for the 1992 Landers earthquake.
Fig. 14. Same as Fig. 12 but for the 1992 Big Bear earthquake.
Fig. 15. Same as Fig. 12 but for the 1994 Northridge earthquake.
235
236
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 225±236
References [1] Blume JA et al. In: San Fernando, California Earthquake of February 9, 1971, vol. 1, Part A, p. 359±393. Washington, DC: US Department of Commerce, National Oceanic and Atmospheric Administration, 1973. [2] Ivanovic SS, Trifunac MD, Todorovska MI. On identi®cation of damage in structures via wave travel times. In: Proc NATO Advanced Research Workshop on Strong Motion Instrumentation for Civil Engineering Structures, Istanbul, 2±5 June, 1999. [3] Ivanovic SS, Novikova EI, Gladkov AA, Todorovska MI. Ambient vibration tests of a seven story reinforced concrete building damaged by the 1994 Northridge, California earthquake. Soil Dynam Earthquake Engng 2000;19:391±411. [4] Ivanovic SS, Trifunac MD, Todorovska MI. Ambient vibration tests of structuresÐa review. Indian Society of Earthquake Technology Journal 2000;37: in press. [5] Lee VW, Trifunac MD. Automatic digitization and processing of accelerograms using PC. Department of Civil Engineering, Report No. 90-03, University of Southern California, Los Angeles, California, 1990. [6] Safak E. Detection of seismic damage in multi-story buildings by using wave propagation analysis. Proc Sixth US National Conf on Earthquake Engineering, Seattle, Washington, 1998. [7] Shakal A, Huang M, Darragh R, Cao T, Sherburne R, Malhotra P, Cramer C, Syndov R, Graizer V, Maldonado G, Peterson C, Wimpole J. CSMIP strong motions records from the Northridge, California, earthquake of 17 January 1994. Report no. OSMS 94-07, California Dept of Conservation, Div of Miner and Geology, Sacramento, California. [8] Todorovska MI, Ivanovic SS, Trifunac MD. Wave propagation in a seven-story reinforced concrete building Part I: theoretical models. Soil Dynam Earthquake Engng 2001;21:211±23. [9] Trifunac MD. A three dimensional dislocation model for the San
[10]
[11]
[12] [13]
[14]
[15] [16] [17] [18]
Fernando, California, earthquake of February 9, 1971. Bull Seism Soc Amer 1974;64:149±72. Trifunac MD, Todorovska MI. Recording and interpreting earthquake response of full-scale structures. Proc NATO Workshop on Strong Motion Instrumentation for Civil Engineering Structures, Istanbul, June 2±5, 1999. Kluwer 1999. Trifunac MD, Todorovska MI. Evolution of accelerographs, data processing, strong motion arrays and amplitude and spatial resolution in recording strong earthquake motion. Soil Dynam Earthquake Engng (submitted for publication). Trifunac MD, Brady AG, Hudson DE. Strong motion earthquake accelerograms, digitized and plotted data, vol. II, Part C, EERL-7251. Pasadena, California: California Institute of Tech, 1973. Trifunac MD, Ivanovic SS, Todorovska MI, Novikova EI, Gladkov AA. Experimental evidence for ¯exibility of a building foundation supported by concrete friction piles. Soil Dynam Earthquake Engng 1999;18:169±87. Trifunac MD, Ivanovic SS, Todorovska MI. Seven story reinforced concrete building in Van Nuys, California: strong motion data recorded between 7 February 1971 and 9 December 1994, and description of damage following the Northridge 17 January 1994 earthquake. Report no. CE 99-02, Dept. of Civil Eng., University of Southern California, Los Angeles, California, 1999. Trifunac MD, Lee VW, Todorovska MI. Common problems in automatic digitization of strong motion accelerograms. Soil Dynam Earthquake Engng 1999;18(7):519±30. Trifunac MD, Ivanovic SS, Todorovska MI. Apparent periods of a building, part I: Fourier analysis. J Struct Engng ASCE 2001;May: in press. Trifunac MD, Ivanovic SS, Todorovska MI. Apparent periods of a building, part II: time-frequency analysis. J Struct Engng ASCE 2001;May: in press. Wald DT, Heaton TH. The slip history of the 1994 Northridge, California, earthquake determined from strong-motion, teleseismic, GPS and leveling data. Bull Seism Soc Amer 1996;86(1B):549±70.
