Mechanism of Collapse, Sensitivity to Ground Motion Features, and Rapid Estimation of the Response of Tall Steel Moment Frame Buildings to Earthquake Excitation

This study explores the behavior of two tall steel moment frame buildings and their variants under strong earthquake ground shaking through parametric analysis using idealized ground motion waveforms. Both fracture-susceptible as well as perfect-connection conditions are investigated. Ground motion velocity waveforms are parameterized using triangular (sawtooth-like) wave-trains with a characteristic period (T), amplitude(peak ground velocity, PGV ), and duration (number of cycles, N). This idealized representation has the desirable feature that the response of the target buildings under the idealized waveforms closely mimics their response under the emulated true ground motion waveforms. A suite of nonlinear analyses are performed on four tall building models subjected to these idealized wave-trains, with T varying from 0.5s to 6.0s, PGV varying from 0.125 m/s to 2.5 m/s, and N taking the values of 1 to 5, and 10. This range of parameters should be adequate to characterize the ground motions that can be expected to occur during earthquakes in the 6-8 magnitude range at some distance (say, > 2km) away from the fault. Databases of peak transient and residual interstory drift ratio (IDR), and permanent roof drift are created for each model. The sensitivity of structural response to T, PGV , and N is studied. Severe dynamic response is induced only in the long-period, large-amplitude excitation regime. Through a simple examination of the energy balance during earthquake shaking, it can be shown that the input excitation energy is small for excitation with periods shorter than the structural period, whereas it is proportional to the square of the ground velocity if the excitation periods are much longer than the structural periods. Thus, collapse-level response can be induced only by long-period, moderate to large PGV ground excitation. The collapse initiation regime expands to lower ground motion periods and amplitudes with increasing number of ground motion cycles. It should be noted that the energy balance analysis is not appropriate for excitation velocities that are extreme where conservation of momentum may be more applicable. However, peak ground velocity from earthquakes seldom exceeds 2.5m/s and energy balance would generally be applicable. The close examination of one instance of collapse shows damage (yielding and/or fracture) localizing in a few stories in the form of a "quasi-shear" band (QSB) comprising of plastic hinges at the top of all columns in the uppermost story of the band, at the bottom of all columns in the lowermost story of the band, and at both ends of all beams in the intermediate stories. Such a pattern of hinging results in shear-like deformation in these stories, resembling plastic shear bands in ductile solids. Most of the lateral deformation due to seismic shaking is concentrated in this band. When the overturning 1st-order and 2nd-order (P - ) moments from the inertia of the overriding block of stories exceed the moment-carrying capacity of the quasi-shear band, it loses stability and collapses. This initiates gravity-driven progressive collapse of the overriding block of stories. Thus, the collapse mechanism initiates as a sidesway mechanism that is taken over by gravity once the quasi-shear band is destabilized. There are Ns(Ns+1) 2 possible quasi-shear bands (and an equal number of sidesway collapse mechanisms) in either principal direction of an Ns-story moment frame building. More than one quasi-shear bands could occur during the entire duration of strong earthquake shaking. The band exhibiting the greatest distress (termed the "primary" quasi-shear band) iv ultimately evolves into a sidesway collapse mechanism. The formation of the quasi-shear band under single-cycle excitations is explained through the classical uniform shear-beam analogy to moment frame buildings. Under low-intensity motions (PGV < 0.25m/s)with periods in the 0.5s-6s range excitation energy is low. As a result, structural response is predominantly elastic and is analogous to that of a uniform elastic shear-beam through which a shear wave propagates. For moderate-intensity excitations (0.25m/s PGV < 1.5m/s), the reverse phase of the incident pulse constructively interferes with the reflected forward phase causing yielding in the region of positive interference,very similar to what would occur in a uniform inelastic shear-beam. The primary quasi-shear band migrates down the building with increasing pulse period. However, this migration slows down with increasing period and gets arrested nominally between floors 3 and 9 for the existing building, and between floors 3 and 8 for the redesigned building, whereas the peak strain in the corresponding inelastic uniform shear-beam continues to migrate to the very bottom. This is a direct result of the non-uniformity of the buildings. Going from the top of the building to the bottom, there is a gradual increase in the strength and stiffness of the structure. The increased strength at the bottom does not allow yielding to permeate into those stories. Now,excitation energy imparted to the structure can be large enough only under long-period ground motion in the context of the target buildings. Therefore, collapse-level response must be accompanied by the formation of the primary quasi-shear band in the vicinity of the stories where the downward migration of the QSB (with increasing T) is arrested. For high-intensity excitations (PGV > 1.5m/s) that are sufficiently long-period, the pulse may yield the structure on its way up the building. The strength of the building drops as the pulse travels up the building. However, inertial forces drop as well, as a result of fewer stories above contributing to the mass. The narrow band of stories with an optimal combination of low-enough strength and high-enough inertial force demand is where peak yielding occurs. This region is identical to the region where the downward migration of the primary quasi-shear band is arrested under moderate-intensity, long-period excitation. This is because the governing factor dictating the location of the band in both cases is strength non-uniformity. As the wave travels up the building, it is reflected off the roof with a change in sign. Because the period of the incident wave is sufficiently long (a necessary condition for large input excitation energy), the reverse phase of the incident pulse constructively interferes with the reflected forward phase causing greatest yielding in the same region as the pre-reflection yielding. To summarize, under both moderate-intensity and highintensity ground motions, input excitation energy large enough to collapse the building requires long-period excitation. Such long-period excitation always causes the formation of the primary quasi-shear band in an optimal set of stories governed by the mass and strength distribution of the building over its height, which are characteristics solely of the structure and not the ground motion. When T and PGV are large enough, it is this band that evolves into a collapse mechanism. This points to the existence of a "characteristic" collapse mechanism or only a few preferred collapse mechanisms (out of the Ns(Ns+1)2 possible mechanisms) in either principal direction of the building. If multiple preferred collapse mechanisms exist, they would be clustered together with significant story-overlap amongst them. The simulations of the four models under idealized ground motion waveforms where collapse occurs do not show the formation of a single (unique) collapse mechanism. However, in each model only one to five v collapse mechanisms occur out of a possible 153 mechanisms in each principal direction of the building. Furthermore, if two or more preferred mechanisms do exist, they have significant story-overlap, typically separated by just one story. For example, the strongly preferred collapse mechanisms in the existing building model (perfect connections) under X direction excitation occur between floors 3 and 9, and floors 4 and 9, while the weakly preferred mechanisms occur between floors 3 and 8, and floors 4 and 8 (four preferred mechanisms out of 153 possible mechanisms, all clustered together within a narrow story zone; two of these mechanisms are in fact a subset of the other two mechanisms). The characteristic and/or preferred collapse mechanisms can be identified by applying the Principle of Virtual Work to all possible quasi-shear bands in a building. Based on plastic analysis principles, the band that is destabilized by the smallest acceleration of the over-riding block of stories is the characteristic collapse mechanism. If one or more bands exist that have destabilizing accelerations close to that of the characteristic collapse band, say within 5%, then these bands may evolve into collapse mechanisms as well. This method identifies all the preferred collapse mechanisms in all four building models satisfactorily. One application of the structural response database built for the sensitivity study is the rapid estimation of structural response immediately following an earthquake if the ground motion records become available. The best fit of the idealized wave-trains in the database to the ground motion record can be determined using the least absolute deviation method. The corresponding key structural response metrics can be extracted from the database using a simple table look-up approach. Such a method, when applied to a suite of nearsource records, predicts peak transient IDR remarkably well. Gaussian mean estimation error on the peak transient IDR is 0.0006, with a standard deviation of 0.0069. A minor modification to this approach is needed when applying it to multi-cycle far-field records. This modified approach is used to estimate the peak transient IDR response of the buildings under synthetic waveforms from a large hypothetical San Andreas fault earthquake. The Gaussian mean error for this estimation is 0.0011, with a standard deviation of 0.0209, slightly worse than for the near-source records, nevertheless within one "performance level" - good enough for emergency response decision-making. The same approach can be used for ball-park estimation of structural response under any given earthquake record, in lieu of comprehensive nonlinear analysis

Hope for the Best, Prepare for the Worst: Response of Tall Steel Buildings to the ShakeOut Scenario Earthquake

This work represents an effort to develop one plausible realization of the effects of the scenario event on tall steel moment-frame buildings. We have used the simulated ground motions with three-dimensional nonlinear finite element models of three buildings in the 20-story class to simulate structural responses at 784 analysis sites spaced at approximately 4 km throughout the San Fernando Valley, the San Gabriel Valley, and the Los Angeles Basin. Based on the simulation results and available information on the number and distribution of steel buildings, the recommended damage scenario for the ShakeOut drill was 5% of the estimated 150 steel moment-frame structures in the 10–30 story range collapsing, 10% red-tagged, 15% with damage serious enough to cause loss of life, and 20% with visible damage requiring building closure

Preparing for the Big One

Approximately 2.75 million deaths have occurred in 3000 earthquakes in the last 105 years between 1900 and 2004 (Figure 1A). About one-half of these occurred in the seven deadliest events, i.e., a few events dominate historical death count. These events did not necessarily have large magnitudes, but occurred close to heavily populated regions. If these not-so-large earthquakes could cause such destruction, one can only imagine what would happen if an extreme event were to occur. An extreme event can be defined as one of large magnitude occurring in the proximity of a densely populated region. Extreme events are rare because large magnitude events are rare. Shown in Figure 1B is the Gutenberg-Richter relation for all earthquakes that have occurred between 1904 and 2000 (Kanamori and Brodsky 2001). In these 96 years, fewer than one magnitude 8.0 earthquake has occurred on average each year. Traditionally, civil engineers have adopted an observe, learn, and improve approach for earthquake damage mitigation. Unfortunately, with extreme events being rare, the learning process is slow and, as a result, corrective measures are ineffective. In fact, we have not seen the effects of a large magnitude earthquake occurring close to heavily populated urban regions such as Los Angeles, Seattle, Istanbul, Jakarta, Tokyo, Taipei, Kaosiung, Delhi, Mumbai, Calcutta, Beijing, etc. in recent years. The recent magnitude 6.7, January 17, 1994, Northridge earthquake, the magnitude 6.9, January 17, 1995, Kobe earthquake, the magnitude 7.4, August 17, 1999, Kocaeli earthquake, and the magnitude 7.7, September 21, 1999, ChiChi earthquake have provided us with glimpses of what we can expect from a major earthquake. But the data from magnitude 8 earthquakes in urban settings is quite limited. Although the magnitude 8.0, September 19, 1985, Michoacan earthquake killed 10000 people and caused significant damage in Mexico City, it was centered more than 360 km away from Mexico City. Both the magnitude 9.5, May 22, 1960, Chile and the magnitude 9.2, March 28, 1964, Prince William Sound, Alaska earthquakes occurred close to sparsely populated regions. The magnitude 7.8, July 28, 1976, Great Tangshan earthquake, the magnitude 8.3, September 1, 1923, Great Kanto earthquake, and the magnitude 7.7, April 18, 1906, San Francisco earthquake provide the best clue to what could be expected from a large earthquake close to an urban center. The fires following the 1923 and 1906 earthquakes destroyed the cities of Tokyo and San Francisco, respectively, although quite a bit of damage can be attributed to ground shaking as well. Ninety percent of the buildings in the city of Tangshan were flattened in the 1976 earthquake. Unfortunately, recorded data from these earthquakes is minimal. As a result, if we are to prepare for an extreme earthquake striking one of our major metropolitan centers, we cannot rely solely on the traditional approach of learning from observations

Performance of Two 18-Story Steel Moment-Frame Buildings in Southern California During Two Large Simulated San Andreas Earthquakes

Using state-of-the-art computational tools in seismology and structural engineering, validated using data from the Mw=6.7 January 1994 Northridge earthquake, we determine the damage to two 18-story steel moment-frame buildings, one existing and one new, located in southern California due to ground motions from two hypothetical magnitude 7.9 earthquakes on the San Andreas Fault. The new building has the same configuration as the existing building but has been redesigned to current building code standards. Two cases are considered: rupture initiating at Parkfield and propagating from north to south, and rupture propagating from south to north and terminating at Parkfield. Severe damage occurs in these buildings at many locations in the region in the north-to-south rupture scenario. Peak velocities of 1 m.s−1 and 2 m.s−1 occur in the Los Angeles Basin and San Fernando Valley, respectively, while the corresponding peak displacements are about 1 m and 2 m, respectively. Peak interstory drifts in the two buildings exceed 0.10 and 0.06 in many areas of the San Fernando Valley and the Los Angeles Basin, respectively. The redesigned building performs significantly better than the existing building; however, its improved design based on the 1997 Uniform Building Code is still not adequate to prevent serious damage. The results from the south-to-north scenario are not as alarming, although damage is serious enough to cause significant business interruption and compromise life safety

On the Modeling of Elastic and Inelastic, Critical- and Post-Buckling Behavior of Slender Columns and Bracing Members

Analyzing tall braced frame buildings with thousands of degrees of freedom in three dimensions subject to strong earthquake ground motion requires an efficient brace element that can capture the overall features of its elastic and inelastic response under axial cyclic loading without unduly heavy discretization. This report details the theory of a modified elastofiber (MEF) element developed to model braces and buckling-sensitive slender columns in such structures. The MEF element consists of three fiber segments, two at the member ends and one at mid-span, with two elastic segments sandwiched in between. The segments are demarcated by two exterior nodes and four interior nodes. The fiber segments are divided into 20 fibers in the crosssection that run the length of the segment. The fibers exhibit nonlinear axial stress-strain behavior akin to that observed in a standard tension test in the laboratory, with a linear elastic portion, a yield plateau, and a strain hardening portion consisting of a segment of an ellipse. All the control points on the stress-strain law are user-defined. The elastic buckling of a member is tracked by updating both exterior and interior nodal coordinates at each iteration of a time step, and checking force equilibrium in the updated configuration. Inelastic post-buckling response is captured by fiber yielding in the nonlinear segments. A user-defined probability distribution for the fracture strain of a fiber in a nonlinear segment enables the modeling of premature fracture, observed routinely in cyclic tests of braces. If the probabilistically determined fracture strain of a fiber exceeds the rupture strain, then the fiber will rupture rather than fracturing. While a fractured fiber can take compression, it is assumed that a ruptured fiber cannot. Handling geometric and material nonlinearity in such a manner allows the accurate simulation of member-end yielding, mid-span elastic buckling and inelastic post-buckling behavior, with fracture or rupture of fibers leading to complete severing of the brace. The element is integrated into the nonlinear analysis framework for the 3-D analysis of steel buildings, FRAME3D. A series of simple example problems with analytical solutions, in conjunction with data from a variety of cyclic load tests, is used to calibrate and validate the element. Using a fiber segment length of 2% of the element length ensures that the elastic critical buckling load predicted by the MEF element is within 5% of the Euler buckling load for box and I-sections with a wide range of slenderness ratios (L/r = 40, 80, 120, 160, and 200) and support conditions (pinned-pinned, pinned-fixed, and fixed-fixed). Elastic post-buckling of the Koiter-Roorda L-frame (tubes and I-sections) with various member slenderness ratios (L/r = 40, 80, 120, 160, and 200) is simulated and shown to compare well against second-order analytical approximations to the solution. The inelastic behavior of struts under cyclic loading observed in the Black et al. and the Fell et al. experiments is numerically simulated using MEF elements. Certain parameters of the model (e.g., fracture strain, initial imperfection, support conditions, etc.) that are not controllable and/or unmeasured during the tests are tuned to realize the best possible fit between the numerical results and the experimental data. A similar comparison is made between numerical results using the MEF element and the experimental data by Tremblay et al. collected from cyclic testing of single-bay braced frames. Finally, a FRAME3D model of a full-scale 6-story braced frame structure that was pseudodynamically tested by the Building Research Institute of Japan subjected to the 1978 Miyagi-Ken-Oki earthquake record, is analyzed iv and shown to closely mimic the experimentally observed behavior. To summarize, the MEF element is able to incorporate all the characteristic features of slender columns and braces that significantly affect their elastic and inelastic, critical and post-buckling behavior, and is remarkably effective in capturing the essence of said behavior, even with the vast uncertainty associated with the buckling phenomenon. To aid in the evaluation of the collapse-prediction capability of competing methodologies, a benchmark problem of a water-tank subjected to the Takatori near-source record from the 1995 Kobe earthquake, scaled down by a factor of 0.32, is proposed. The water-tank is so configured as to have a unique collapse mechanism (under all forms of ground motion), of overturning due to P - instability resulting from column and brace buckling at the base. A FRAME3D model of the tank reveals severe buckling in the bottom megacolumns on the west face of the tower, followed almost instantaneously by compression brace buckling on the north and south faces, when the structure is hit by the Takatori near-source pulse, resulting a tilt in the structure. Subsequent shaking induces P - instability resulting in complete collapse of the tank

Desorption Kinetics of O and CO from Graphitic Carbon Surfaces

The desorption of O/CO from graphitic carbon surfaces is investigated using a one-dimensional model describing the adsorbate interactions with the surface phonon bath. The kinetics of desorption are described through the solution of a master equation for the time-dependent population of the adsorbate in an oscillator state, which is modified through thermal fluctuations at the surface. The interaction of the adsorbate with the surface phonons is explicitly captured by using the computed phonon density Of states (PDOS) of the surface. The coupling of the adsorbate with the phonon bath results in the transition of the adsorbate up and down a vibrational ladder. The adsorbate-surface interaction is represented in the model using a Morse potential, which allows for the desorption process to be directly modeled as a transition from bound to free (continuum) state. The PDOS is a property of the material and the lattice; and is highly sensitive to the presence of defects. The effect of etch pits along with random surface defects on the PDOS is considered in the present work. The presence of defects causes a redshift and broadening of the PDOS, which in turn changes the phonon frequency modes available for adsorbate coupling at the surface. Using the realistic PDOS distributions, the phonon-induced desorption (PID) model was used to compute the transition and desorption rates for both pristine and defective systems. Mathissens rule is used to compute the phonon relaxation time for pristine and defective systems based on the phonon scattering times for each of the different scattering processes. First, the desorption rates of the pristine system is fitted against the experimental values to obtain the Morse potential parameters for each of the observed adatoms. These Morse potential parameters are used along with the defective PDOS and phonon relaxation time to compute the desorption rates for the defective system. The defective system rates (both transition and desorption) were consistently lower in comparison with the pristine system. The difference between the transition rates is more significant at lower initial states due to higher energy spacing between the levels. In the case of the desorption rates, the difference between the defective and pristine system is more significant at higher temperatures. The desorption rates for each of the system shows an order of magnitude decrease with the strongly bound systems exhibiting the greatest reduction in the desorption rates

Performance of 18-Story Steel Momentframe Buildings during a large San Andreas Earthquake - A Southern California-Wide End-to-End Simulation

The mitigation of seismic risk in urban areas in the United States and abroad is of major concern for all governments. Unfortunately no comprehensive studies have attempted to address this issue in a rigorous, quantitative manner. This study tackles this problem head-on for one typical class of tall buildings in southern California. The approach adopted here can be used as a template to study earthquake risk in other seismically sensitive regions of the world, such as Taiwan, Japan, Indonesia, China, South American countries (Chile, Bolivia, etc.), and the west coast of the United States (in particular, Seattle). In 1857 a large earthquake of magnitude 7.9 [1] occurred on the San Andreas fault with rupture initiating at Parkeld in Central California and propagating in a southeasterly direction over a distance of more than 360 km. Such a unilateral rupture produces signicant directivity toward the San Fernando and Los Angeles basins. Indeed, newspaper reports (Los Angeles Star [2, 3]) of sloshing observed in the Los Angeles river point to long-duration (1-2 min) and long-period (2-8 s) shaking, which could have a severe impact on present-day tall buildings, especially in the mid-height range. To assess the risk posing tall steel moment-frame buildings from an 1857-like earthquake on the San Andreas fault, a nite source model of the magnitude 7.9 November 3, 2002 Denali fault earthquake is mapped on to the San Andreas fault with rupture initiating at Parkeld in Central California and propagating a distance of about 290 km in a south-easterly direction. As the rupture proceeds down south from Parkeld and hits the big bend on the San Andreas fault, it sheds off a signicant amount of energy into the San Fernando valley, generating large amplitude ground motion there. A good portion of this energy spills over into the Los Angeles basin with many cities along the coast such as Santa Monica and Seal Beach and more inland areas going east from Seal beach towards Anaheim experiencing long-duration shaking. In addition, the tail-end of the rupture sheds energy from SH/Love waves into the Baldwin Park-La Puente region, which is bounded by a line of mountains that creates a mini-basin, further amplifying the ground motion. The peak velocity is of the order of 1 m.s in the Los Angeles basin, including downtown Los Angeles, and 2 m.s in the San Fernando valley. Signicant displacements occur in the basins but not in the mountains. The peak displacements are in the neighborhood of 1 m in the Los Angeles basin and 2 m in the San Fernando valley. The ground motion simulation is performed using the spectral element method based seismic wave propagation program, SPECFEM3D. To study the effects of the ground motion simulated at 636 sites (spread across southern California, spaced at about 3.5 km each way), computer models of an existing 18-story steel moment-frame building and a redesigned building with the same conguration (redesigned to current standards using the 1997 Uniform Building Code) are analyzed using the nonlinear structural analysis program, FRAME3D. For these analyses, the building Y direction is aligned with the geographical north direction. As expected, the existing building model fares much worse than the redesigned building model. Fracture occurs in at least 25% of the connections in this building when located in the San Fernando valley. About 10% of connections fracture in the building when located in downtown Los Angeles and the mid-Wilshire district (Beverly Hills), while the numbers are about 20% when it is located in Santa Monica, west Los Angeles, Inglewood , Alhambra, Baldwin Park, La Puente, Downey, Norwalk, Brea, Fullerton, Anaheim and Seal Beach. The peak interstory drifts in the middle-third and bottom-third of the existing building are far greater than the top-third pointing to damage being localized to the lower oors. The localization of damage in the lower oors rather than the upper oors could potentially be worse because of the risk of more oors pancaking on top of each other if a single story gives way. Consistent with the extent of fracture observed, the peak drifts in the existing building exceed 0.10 when located in the San Fernando valley, Baldwin Park and neighboring cities, Santa Monica, west Los Angeles and neighboring cities, Norwalk and neighboring cities, and Seal Beach and neighboring cities, which is well into the postulated collapse regime. When located in downtown Los Angeles and the mid-Wilshire district, the building would barely satisfy the collapse prevention criteria set by FEMA [4] with peak drifts of about 0.05. The performance of the newly designed 18-story steel building is signicantly better than the existing building for the entire region. However, the new building still has signicant drifts indicative of serious damage when located in the San Fernando valley or the Baldwin Park area. When located in coastal cities (such as Santa Monica, Seal Beach etc.), the Wilshire-corridor (west Los Angeles, Beverly Hills, etc.), the mid-city region (Downey, Norwalk, etc.) or the booming Orange County cities of Anaheim and Santa Ana, it has peak drifts of about 0.05, once again barely satisfying the FEMA collapse prevention criteria [5]. In downtown Los Angeles it does not undergo much damage in this scenario. Thus, even though this building has been designed according to the latest code, it suffers damage that would necessitate closure for some time following the earthquake in most areas, but this should be expected since this is a large earthquake and building codes are written to limit the loss of life and ensure "collapse prevention" for such large earthquakes, but not necessarily limit damage. Unfortunately, widespread closures such as this could cripple the regional economy in the event of such an earthquake. A second scenario considered in the study involves the same Denali earthquake source mapped to the San Andreas fault but with rupture initiating in the south and propagating to the north (with the largest amount of slip occurring to the north in Central California) instead of the other way around. The results of such a scenario indicate that ground shaking would be far less severe demonstrating the effects of directivity and slip distribution in dictating the level of ground shaking and the associated damage in buildings. The peak drifts in existing and redesigned building models are in the range of 0.02-0.04 indicating that there is no signicant danger of collapse. However, damage would still be signicant enough to warrant building closures and compromise life safety in some instances. The ground motion simulation and the structural damage modeling procedures are validated using data from the January 17, 1994, Northridge earthquake while the band-limited nature of the ground motion simulation (limited to a shortest period of 2 s by the current state of knowledge of the 3-D Earth structure) is shown to have no signicant effect on the response of the two tall buildings considered here with the use of observed records from the 1999 Chi Chi earthquake in Taiwan and the 2001 Tokachi-Oki earthquake in Japan

Sensitivity of the Earthquake Response of Tall Steel Moment Frame Buildings to Ground Motion Features

The seismic response of two tall steel moment frame buildings and their variants is explored through parametric nonlinear analysis using idealized sawtooth-like ground velocity waveforms, with a characteristic period (T), amplitude (peak ground velocity, PGV), and duration (number of cycles, N). Collapse-level response is induced only by long-period, moderate to large PGV ground excitation. This agrees well with a simple energy balance analysis. The collapse initiation regime expands to lower ground motion periods and amplitudes with increasing number of ground motion cycles