THE USE OF POWER FOR THE CHARACTERIZATION OF EARTHQUAKE GROUND MOTIONS

Transcription

1 13 th World Conference on Earthquake Enineerin Vancouver, B.C., Canada Auust 1-6, 004 Paper No. 574 THE UE OF POWER FOR THE CHARACTERIZATION OF EARTHQUAKE GROUND MOTION Kenneth ELWOOD 1 and Erik NIIT UMMARY Power, defined as the time rate of chane of enery, has received little or no attention in the literature as a useful parameter for selectin critical round motions for seismic desin. Power can be used to distinuish between distant and near-fault round motions which may have approximately the same maximum input enery. Furthermore, the rate at which enery is dissipated throuh either dampin or yieldin of the structure can be determined by calculatin the dampin and hysteretic power terms. This study investiates the characteristics of power time histories for selected round motions and an elasticperfectly-plastic DOF oscillator. Power spectra are also developed to investiate the variation of the maximum input power with the period of the structure and the assumed hysteretic model. It is noted that short period structures tend to experience a constant input power reardless of the ductility demand. INTRODUCTION The use of enery concepts in earthquake-resistant desin was first proposed by Housner [1]; however, little research has done in this area until the past 0 years. Uan and Bertero [] proposed the use of enery spectra, developed from sinle deree of freedom (DOF) systems, for earthquake-resistant desin and proposed methods of evaluatin the enery capacity of structural materials. To date, the majority of research has been directed toward correlatin the input enery to the severity of the earthquake round motion and the development of damae indices based, in part, on hysteretic enery demand. Thus far, no sinificant research has been undertaken to study the rate of enery input into the structure (i.e. input power) and the rate at which enery is balanced by the various storae and dissipation mechanisms in the structure (i.e. hysteretic, dampin, kinetic, and strain power). ince pulse-type and lon-duration round motions can impart the same total input enery to a system, while requirin the enery to be dissipated at different rates, it may be beneficial to quantify and characterize the earthquake power when selectin critical round motions. Althouh not considered explicitly in calculations, the concept of power has been discussed by Housner and Jennins [3]: If enery is fed to a structure at a sufficiently slow rate [i.e. low power], the dissipation due to dampin will prevent the structural members from becomin overstressed, but durin a hih rate of enery input [i.e. hih power] the dissipation by dampin may be 1 Assistant Professor, Department of Civil Enineerin, University of British Columbia, Vancouver, Canada Graduate tudent, Department of Civil Enineerin, University of British Columbia, Vancouver, Canada

2 inadequate to hold the enery level in the structure below the threshold of damae, and then plastic deformation, crackin, etc. may result. The objective of this paper is thus to characterize near-fault (or pulse-type) round motions and londuration round motions, usin the concept of earthquake power, and to a lesser extent, earthquake enery. This will be accomplished throuh a description of the characteristics of power time histories and power spectra. Earthquake power will be introduced for an elastic-perfectly-plastic (EPP) DOF oscillator subjected to a series of recorded pulse-type and lon-duration round motions. The influence of the assumed hysteretic model will also be investiated. This paper will present the use of power demand as a measure of the severity and damae potential of earthquakes. In this context, power is viewed as the rate of chane of enery and should not be confused with the power spectral density, which relates to the frequency content of an earthquake round motion. ENERGY AND POWER EQUATION Enery The relative enery balance equation for a DOF oscillator is determined by interatin the equation of motion with respect to relative displacement, u, Uan and Bertero []. m udu & + cudu & + f du = mu& du (.1) u u For Eq. (.1), m is the mass, c is the coefficient of viscous dampin, f is the restorin force, u& & is the round acceleration, and u, u&, and u& & are the relative displacement, velocity, and, acceleration of the mass, respectively. It should be noted that the restorin force, f represents the sprin force of the inelastic system. From the left to the riht side of Eq. (.1), the first term is referred to as the relative kinetic enery E K, the second term is the dampin enery, E D, and the third term represents the absorbed enery and consists of the elastic strain enery, E, and the inelastic hysteretic enery, E H. The term on the riht-hand side of Eq. (.1) corresponds to the relative input enery of the round motion, E I, or the work done by the effective earthquake force, m u&. Hence, the enery balance equation can expressed as follows, K D u H u E + E + E + E = E (.) For a non-stiffness deradin system, the strain enery can be expressed as: ( f ) E = (.3) k where k is the initial elastic stiffness of the system. For an EPP hysteretic model, the hysteretic enery can be expressed concisely as the product of the yield force level, F Y, and the total cumulative plastic cum displacements,, p I E = (.4) H F Y The differences between the absolute and relative formulations of the enery balance equation are discussed in detail by Uan and Bertero []. This study will concentrate on the relative formulation for two reasons. First, relative enery terms may be considered more useful since the sprin and dampin forces in a structure are related to the relative displacements and velocities. econd, for lon period structures the absolute input enery will approach zero, and hence, is not useful for the desin of such cum p

3 structures. For very short period structures the relative input enery will approach zero; however, these structures require a strenth-based desin to avoid excessive ductility demands, and hence, enery concepts may be inappropriate. Thus, for the remainder of this paper any reference to input enery and kinetic enery will refer to the relative input and kinetic eneries. Power Power is physically defined as the time rate of chane of enery. The relative power equation overnin the motion of the DOF oscillator may be formulated by differentiatin Eq. (.1) with respect to time: d d d d mudu & + cudu & + f du = mu&& du (.5) dt u dt u dt u dt u Carryin out the interation above, muu &&& + cu& + f u& = mu&& u& (.6) imilar to the enery terms expressed in Eq. (.), the terms on the left-hand side of the equation correspond to the relative kinetic power, P K, the dampin power, P D, and the absorbed power which consists of the elastic strain power, P, and the inelastic hysteretic power, P H. The term on the riht-hand side of the equation defines the relative input power of the round motion, P I. As with the enery balance, the power balance can be rewritten as: P + P + P + P = P (.7) K D Note that the strain and hysteretic power terms for an EPP system can be calculated directly by differentiation of Eq. (.3) and Eq. (.4), respectively, d dt ( f ) P = = k d d d P = E = ( F ) = F ( ) (.9) H H Y cum Y cum dt dt dt The spectral analysis proram Bispec [4] was used throuhout this study to calculate the numerical results. H f k d dt I ( f ) ELECTION OF GROUND MOTION For this study, two broad classes of round motions have been considered: near-fault records from crustal earthquakes to represent pulse-type round motions (Table 1), and records from subduction-type earthquakes to represent lon-duration round motions (Table ). All round motions were selected from records used for the AC teel Project [5] and have a return period of approximately 475 years. (.8)

4 Table 1. Pulse-type round motions [5]. AC Name Earthquake tation Manitude cale Distance Duration 1 Year (km) (s) NF01 Tabas NF03 Loma Prieta Los Gatos NF05 Loma Prieta Lex. Dam NF07 C. Mendocino Petrolia NF09 Erzincan NF11 Landers NF13 Northride Rinaldi NF15 Northride Olive View NF17 Kobe Kobe NF19 Kobe Takatori Duration calculated based on bracketed duration for 5 [6]. Table. Lon-duration round motions [5]. AC Earthquake tation Manitude cale Distance Duration 1 Year Name (km) (s) E05 West Olympia Washinton E06 West Olympia Washinton E07 West eattle Washinton Army Base E08 West eattle Washinton Army Base E15 East Tacoma Washinton County E16 East Tacoma Washinton County E17 Chile Llolleo E18 Chile Llolleo E19 Chile Vina del Mar E0 Chile Vina del Mar Duration calculated based on bracketed duration for 5 [6]. ENERGY AND POWER TIME HITORIE Enery and power time histories were produced for an DOF oscillator with a period of seconds and 5% viscous dampin. The non-linear characteristics of the system were modeled usin an EPP hysteretic model with a yield force level defined by a normalized strenth of unity. (The normalized strenth is η = m PGA, where PGA refers to the peak round acceleration). iven by ( ) F Y

5 Enery Normalized to Peak Input Enery 0. E I E K E D Enery Normalized to Peak Input Enery 0. E E H Power Normalized to Peak Input Power P I P K P D Power Normalized to Peak Input Power Fiure 1. Non-linear enery and power time history for NF17 round motion (Kobe, 1995) (T = sec, ζ = 5%). P P H

6 Enery Normalized to NF17 Peak Input Enery Power Normalized to NF17 Peak Input Power 0. E I E K E D P I P K P D Enery Normalized to NF17 Peak Input Enery Power Normalized to NF17 Peak Input Power 0. E E H Fiure. Non-linear enery and power time history for E19 round motion (Vina del Mar, 1985) (T = sec, ζ = 5%). (Peak input enery equal to input enery for NF17 round motion. Power terms normalized to peak input power for NF17 round motion) P P H

7 It should be noted that the observations presented may depend on the chosen characteristics of the particular systems, in particular the period of the DOF, the hysteretic model, and round motions evaluated. The influence of such characteristics will be discussed in subsequent sections on enery and power spectra. The enery and power time histories for the NF17 pulse-type round motion and the E19 lon-duration round motion are presented in Fiures 1 and. The acceleration amplitude of the E19 round motion was scaled by 5 to achieve the peak input enery of the NF17 round motion. To facilitate comparison of the results, the enery and power time histories for both round motions were normalized to the peak input enery and the peak input power for the NF17 round motion. It should be noted that the response of both systems was computed with a normalized strenth of ; however the PGA of the NF17 round motion was nearly twice that of the E19 round motion, and, for the same mass, the NF17 yield strenth was thus nearly twice that of the E19 yield strenth. The NF17 and E19 enery time histories illustrate some of the characteristic differences between pulse and lon duration round motions. For example, the E19 response requires approximately forty-five seconds, nine times loner than the NF17 response, to input the same amount of enery into the system. The proportion of the input enery dissipated by hysteretic enery is much larer for the NF17 response than for the E19 response (in which the dampin and hysteretic enery increase at rouhly the same rate). The hysteretic enery developed in the E19 round motion is built up over a much larer number of yieldin events than in the NF17 response. Intuitively, the NF17 round motion must impart the same amount of input enery as the E19 round motion over a much shorter time thus requirin a faster rate of enery delivery (i.e. hiher power). The NF17 and E19 power time histories indicate that the peak value of the E19 input power is much smaller than the peak value of the NF17 input power. Indeed, all of the power terms in the E19 power time history are approximately 5% of their value in the NF17 power time history. It should be noted, however, that the P H /P I ratio is only slihtly hiher, while the P K /P I and P /P I are actually slihtly lower for the NF17 response terms when compared with the E19 response. This suests that each power term maintains an approximate proportion of the input power reardless of the round motion, an observation that will be discussed further below. The power time history for the NF17 round motion, shown in Fiure 1, can be used to illustrate several interestin characteristics of the individual power terms. From the lower riht plot of Fiure 1 it is clear that only one of the hysteretic and strain power terms is active at any one time; at the moment of yieldin, the strain power drops to zero while the hysteretic power spikes to the same power manitude as the strain power term immediately prior to yieldin. The variation in time of the strain and hysteretic power terms can be readily understood by considerin the sum of the strain and hysteretic power terms, referred to herein as the sprin power. The kinetic power term and the sprin power term are out of phase by approximately 180. The dampin power, which is relatively small, is out of phase with the kinetic and sprin terms by approximately 90. A neative increase in the strain power term corresponds to unloadin of the DOF elastically. It is also interestin to note from Fiure 1 that all of the power terms convere to zero at the same time. This observation can be explained by notin that each power term in Eq. (.6) is a function of the velocity. Thus, if the velocity approaches zero, all of the power terms must also approach zero. The dampin power will always maintain a positive value due to the dependence on the square of the velocity. While the sprin power is directly related to the velocity and must chane sins when the velocity chanes sins, the hysteretic power, accordin to Eq. (.9), will always be positive since the time derivative of the cumulative plastic displacements is also always positive. The hysteretic power will be

8 zero when the DOF unloads elastically at a peak in the displacement, and hence, at zero velocity. Thus, the sprin power and hysteretic power will both be zero when the velocity is zero, thereby requirin the strain power to also be zero. Also observed from Eq. (.6), the input power is directly related to the round acceleration and must necessarily become zero when the round motion ceases. The acceleration records of five round motions were scaled such that the peak input enery matched that of the NF17 response for three normalized strenth values (with T = sec). The relationship between the peak input power for each round motion, at a iven normalized strenth, and the peak input power of the NF17 response is shown in Table 3. The results suest some dependence of input power on the round motion. The input power of the lon duration round motions (E07, E17, and E19) is consistently less than the input power of the pulse-type round motions (NF13, NF17, and NF19) reardless of the normalized strenth. This observation suests that hih values of input power may characterize pulse-type round motions, while low values of input power may characterize lon-duration round motions. The results in Table 3 further suest that the normalized strenth influences the manitude of P /P I, P H /P I, and P K /P I. The proportion P D /P I, however, appears to be rouhly independent of the normalized strenth and the characteristics of the round motion. Further analyses indicate that, reardless of period, P D /P I is approximately constant for ductility demands reater than. It is also worthy to note that, with the exception of P K /P I for E17, the peak kinetic, strain, hysteretic, and dampin power values (expressed as a fraction of the peak input power) appear to be relatively independent of the selected round motion. Table 3. Peak power terms for six round motions at a constant system period of s and a specified normalized strenth. Normalized trenth 0. 1 Ground cale P I /P I,NF17 P K /P I P /P I P H /P I P D /P I Motion Factor NF NF NF E E E NF NF NF E E E NF NF NF E E E ENERGY AND POWER REPONE PECTRA The previous sections investiated the power and enery response of an arbitrarily selected elasticperfectly-plastic DOF oscillator with a period of seconds. Power and enery response spectra are

9 discussed in the followin sections to investiate the dependence of the maximum response on the oscillator period. The discussion presented herein is limited to the input terms; investiation of hysteretic power and enery spectra can be found elsewhere [7]. Input enery and input power spectra It is convenient to express the input enery and input power as a dimensionless quantity. To achieve this, both terms are normalized by the input enery and input power for lon period structures since both lon period terms convere to a specific value dependin only on the properties of the round motion and the system mass. Recallin that the relative displacement approaches the round displacement for lon period structures, the lon period input enery can be formulated as follows: 1 E ( T ) = mu& du = mu&& du = mu& ( t) (5.1) I u u Thus, the maximum lon period input enery is iven by, 1 E I ( T ) = mpgv,max (5.) where PGV is the peak round velocity. imilarly, the lon period input power can be formulated, from Eq. (.6), as P I ( T ) = mu& u& = m( u&& u& ) Finally, the maximum lon period input power is iven by, P (5.3) (&& u& ) max ( T ) = m u I, max (5.4) Note that Eq. (5.4) represents the maximum value of the product of the round acceleration and round velocity, not the product of peak round acceleration, PGA, and the PGV. ince the lon period input enery and input power are dependent only on the characteristics of the round motion, their independence of system strenth, hysteretic model, and other system specific parameters recommends their use as a means of quantifyin the properties of round motions. Table 4 presents the lon period input enery and input power for the round motions listed in Tables 1 and. Note that the lon period input enery and lon period input power are much larer for the pulse-type round motions than for the lon-duration round motions, likely as a result of the larer round acceleration and velocity amplitude characteristic of near fault round motion. Table 4. Lon period input enery and input power for near fault and subduction event round motions. E I (kn m) P I (kn m/s) E I (kn m) P I (kn m/s) NF E NF E NF E NF E NF E NF E NF E NF E NF E NF E

10 Linear Normalized Input Power Fiure 3: Averaed linear input power and individual round motion input power for (a) pulse-type round motions and (b) lon-duration round motions Normalized Input Enery µ = 1 µ = µ = 4 µ = 8 Normalized Input Power µ = 1 µ = µ = 4 µ = Fiure 4. Constant ductility input enery and power spectra for pulse-type round motions. Input enery and input power spectra, normalized to the input enery and input power at lon periods, were enerated for the twenty round motions presented in Table 1 and Table, above. The normalized near fault input enery and input power spectra were subsequently averaed toether to produce a data set characteristic of pulse-type round motions in eneral, and likewise for the lon-duration round motions. Fiure 3 illustrates the variability between the individual linear input power spectra and the averae spectra for both pulse-type and lon-duration round motions. The averaed input enery and input power spectra for the pulse-type round motions are presented in Fiure 4. Both the non-linear input enery and input power appear to convere to the linear input enery and input power at short periods; however, this converence is much more pronounced for the input power and occurs, accordin to Fiure 4, for all periods less than 6s. From the individual round motion spectra (not shown), the input power converence is more pronounced for some round motions than for others and the period at which diverence occurs varies from approximately 0.3 sec to sec. A similar converence was observed for the linear and non-linear averaed spectral velocity, normalized to PGV, over approximately the same period rane. Recallin Eq. (.6), it is noted that the input power and system velocity are directly related.

11 µ = µ = 4 µ = µ = µ = 4 µ = 8 λ γ Fiure 5. Monotonic hysteretic enery (λ) and normalized displacement (γ) for pulse-type round motions Fiure 5 illustrates the applicability of equal enery and equal displacement rules [7] for the selected pulse-type round motions. hown on the left is the normalized monotonic enery for an EPP system, λ: FYu FY k λ = (5.5) k inelastic max fe where f e is the maximum force level of the equivalent linear system and u inelastic max is the peak inelastic displacement. The equal enery rule is satisfied when λ. hown on the riht of Fiure 5 is the inelastic normalized displacement, γ, defined as the quotient of the peak inelastic displacement, u max, and the elastic peak elastic displacement of the equivalent linear system, u max : inelastic u max γ = (5.6) elastic u The normalized displacement, γ, will approach unity when the equal displacement rule is applicable. Comparin Fiures 4 and 5, the non-linear and the linear input power are converent durin the same period rane (0.1 sec to 6 sec) over which λ diveres dramatically. Note that λ converes to (i.e. equal enery ) at a period of approximately 6 sec, and that the linear and non-linear input power terms divere for periods reater than 6 sec. Furthermore, the fiures suest that the equal enery rule has only a small rane of applicability for pulse-type round motions and that linear input power, or linear spectral velocity, may be a better means of characterizin the non-linear response of near fault round motions at shorter periods. Note also that γ approaches unity beyond a period of sec, supportin the equal displacement rule for lon period structures. In contrast, the averaed input power for lon-duration round motions, Fiure 6, displays converence to the linear input power up to a period of only 0. sec. Fiure 7, showin λ and γ for the lon-duration round motions, suests a converence to λ = at a period of 0. sec. Note also that λ remains closer to unity for a broader period rane (0. sec to sec) than for the pulse-type round motions. This suests that the equal enery principle may be satisfied for lon-duration round motions over a larer period rane than for pulse-type round motions. Fiure 7 further suests that the equal displacement rule (γ ) holds for periods reater than approximately sec. max

12 Normalized Input Power.0 µ = 1 µ = µ = 4 µ = Fiure 6. Averaed, normalized constant ductility input power spectra for lon-duration round motions µ = µ = 4 µ = µ = µ = 4 µ = 8 λ γ Fiure 7. Monotonic hysteretic enery (λ) and normalized displacement (γ) for lon-duration round motions In summary, for the limited round motions considered in this study, the nonlinear DOF response can be approximated by different linear response parameters dependin on the period rane and the eneral type of round motion as shown in Table 5. Further study of these characteristics is required to determine if similar trends are observed for a larer database of round motions. Table 5. ummary of period ranes for different approximations to nonlinear response based on linear response. Type of Equal Max Equal Max Equal Max Ground Motion Input Power Enery Displacement Pulse-type T < 6 sec 6 sec T < sec T sec Lon-duration T < 0. sec 0. sec T < sec T sec

13 ensitivity to Hysteretic Model To investiate the sensitivity of the power quantities to the assumed hysteretic model, analyses were conducted with a bilinear model with 10% strain hardenin and a Clouh-type stiffness deradin model [8]. Fiure 8 suests that the input power is relatively insensitive to the hysteretic model, since the input power spectra for each model is nearly identical, both in terms of salient characteristics and manitude. The Clouh hysteretic model appears to have the effect of smoothin out the input power spectra, but overall, the spectra correlate very well. Note that the input power spectra still convere for periods less than 6 sec, reardless of the assumed hysteretic model. imilarity between the results for the three hysteretic models suests that the conclusions from the previous sections still hold for the bilinear and Clouh hysteretic models. Normalized Input Power EPP (µ = ) H (µ = ) Clouh (µ = ) EPP (µ = 4) H (µ = 4) Clouh (µ = 4) Fiure 8. Input power spectra for three hysteretic models (EPP = Elastic perfectly plastic; H = 10% train hardenin; Clouh = Clouh-type stiffness deradin) CONCLUION This study hihlihts the importance of evaluatin earthquake power demands in conjunction with earthquake enery demands. Generally, the input power demand was observed to depend on the nature of the round motion, and in particular, pulse-type round motions appear to be characterized by hiher input power demand than lon-duration round motions for a constant input enery. The peak value of the strain, hysteretic, kinetic, and dampin power terms normalized by the peak input power appear to be relatively independent of the characteristics of a iven round motion. However, since the input power is dependent on the round motion, the absolute value of the power terms is also on the round motion. The lon period peak input enery and input power depend only on the round velocity and round acceleration and thus provide another means of characterization. It was observed that the lon period input enery and input power for pulse-type round motions tend to be considerably larer than for londuration round motions. The inelastic input power was observed to convere to the linear input power for short period structures. The period rane over which the non-linear and linear input power terms convere was observed to be larer for the pulse-type round motions, compared with the lon-duration round motions. Additionally, it was noted that neither the equal enery nor equal displacement principle were valid within the period

14 rane of converence for the input power. Further study is required to confirm if the above observations are supported by a larer sample of round motions. REFERENCE 1. Housner, G.W., 1956, Limit Desin of tructures to Resist Earthquakes, Proceedins. of the 1956 World Conference on Earthquake Enineerin. Earthquake Enineerin Research Institute: Oakland, California.. Uan, C.-M. and Bertero, V.V., 1988, Use of Enery as a Desin Criterion in Earthquake Resistant Desin, UCB/EERC-88/18. University of California: Berkeley, California. 3. Housner, G.W. and Jennins, P.C., 1977, The capacity of extreme round motions to damae structures, tructural and Geotechnical Mechanics (ed. W.J. Hall), Prentice Hall, Enlewood Cliffs, NJ, p Hachem, M., 1999, Bispec: Interactive Computer Proram for Computation of Bidirectional Nonlinear pectra," NIEE oftware Library: Berkeley, Calif., ( 5. omerville, P.G., 1997, Develop uites of Time Histories Draft Report, AC Joint Venture teel Project. 6. Bolt, B.A., 1969, Duration of stron round motion, Proceedins. of the Fourth World Conference on Earthquake Enineerin. antiao, Chile, pp Elwood, K.J., and Niit, E.J., 004, The Use of Power and Enery in The Characterization of Earthquake Ground Motions, Journal of Earthquake Enineerin (submitted for review). 8. Newmark, N.M. and Hall, W.J., 198, Earthquake pectra and Desin, Monoraph, Earthquake Enineerin Research Institute. 9. Clouh, R.W., 1966, Effect of stiffness deradation on earthquake ductility requirements, Report 66-16, tructural and Materials Research, tructural Enineerin Laboratory, University of California, Berkeley.