The Estimation of Peak Ground-motion Parameters from Spectral Ordinates

Transcription

1 Journal of Earthquake Engineering ISSN: (Print) X (Online) Journal homepage: The Estimation of Peak Ground-motion Parameters from Spectral Ordinates Edmund Booth To cite this article: Edmund Booth (27) The Estimation of Peak Ground-motion Parameters from Spectral Ordinates, Journal of Earthquake Engineering, :, 3-32, DOI:.8/ To link to this article: Published online: 25 May 27. Submit your article to this journal Article views: 263 View related articles Citing articles: 2 View citing articles Full Terms & Conditions of access and use can be found at Download by: [ ] Date: 3 December 26, At: 2:2

2 Journal of Earthquake Engineering, :3 32, 27 Copyright A.S. Elnashai & N.N. Ambraseys ISSN: print / X online DOI:.8/ X Journal UEQE of Earthquake Engineering, Vol., No., December 26: pp. 25 The Estimation of Peak Ground-motion Parameters from Spectral Ordinates Estimating E. Booth Peak Motions from Spectral Ordinates EDMUND BOOTH Edmund Booth Consulting Engineer, Walthamstow, London, UK Relationships are derived between peak ground-motion parameters and the peaks in the corresponding 5% damped response spectrum. The relationships are based on a study of the smoothed spectra from a database of over, horizontal ground-motion records. A more consistent, but more complex, relationship is also derived between peak ground velocity (PGV) and an integration of the 5% damped spectrum, using random vibration theory. A recent proposal to estimate PGV from the.5 sec spectral ordinate is found to give less accurate results than the methods derived in this study, when compared with values derived from the strong ground-motion database; in particular, it consistently underestimates PGV for records with unusually long or short predominant periods. However, when compared with predictive equations for PGV, this method gives results comparable to the other two and is also much the simplest. Relationships between spectral ordinates and PGV are useful for the purpose of estimating PGV from a smoothed design spectrum or from a spectrum derived from ground-motion equations, for uses including estimation of structural damage and geotechnical effects. The relationships between response spectrum peaks and peak groundmotion parameters may also be useful in the selection of real time histories compatible with a smoothed design spectrum. Keywords PGV; PGD; PGA; Spectral Peak; Smoothed Spectrum. Introduction In a recent paper, Bommer and Alarcon [26] discussed the importance of peak ground velocity (PGV) in earthquake engineering. They observed that there is a shortage of ground-motion equations predicting PGV, compared to those for peak ground acceleration (PGA) and spectral ordinates; therefore the need may arise to derive PGV from spectral values. Bommer and Alarcon showed that the relationship between 5% damped spectral response at.5 sec and PGV is more stable with respect to magnitude and has less scatter than is provided by spectral response at other periods, including. sec, which is often used to derive PGV. However, the relationship is subject to significant uncertainty, and Bommer and Alarcon do not advise its use when near-fault directivity effects may be present. More generally, they recommend direct use of a PGV predictive equation wherever possible, and urge that more such equations should be produced in future. The present article reports a study which attempted to improve the reliability of predicting PGV from 5% damped spectral ordinates and to remove some of the shortcomings of basing the relationship on spectral response at a single, fixed period. Two methods are derived; a simple relationship between PGV and peak pseudo spectral velocity and a more complex method based on random vibration theory, which is nevertheless easily Received 2 February 25; accepted 5 June 26. Address correspondence to Edmund Booth, Edmund Booth Consulting Engineer, 67 Orford Road, Walthamstow, London, E7 9NJ, UK. Edmund@Booth-seismic.co.uk 3

3 4 E. Booth implementable in a spreadsheet. The study also investigated the relationship between peak spectral values and peak ground displacement (PGD) or acceleration (PGA), using similar methods. In the rest of the paper, spectral accelerations and velocities are taken as pseudospectral values, without further clarification. 2. Relationship between Peak Spectral Values and Peak Ground-motion Parameters 2. Theoretical Basis There is a clear and simple link between PGA and spectral values; PGA is of course exactly equal to the spectral acceleration (independently of damping level) at zero period. No such rigorous theoretical link exists for PGV or PGD; although for low damping, PGD approximates to the spectral value at very long periods, the periods involved are longer than are normally available. However, there are some grounds for supposing that PGV and PGD might be related to the corresponding peak spectral values. For a single frequency ground-motion of infinite duration, such a theoretical link does exist: peak spectral values (acceleration, velocity or displacement) are related to peak ground parameters by the wellknown ratio /(2ξ), where ξ is the damping ratio. Thus for 5% damping, an infinitely long single frequency ground-motion would produce 5% damped spectral peaks of times the ground-motion peaks, and this applies equally to acceleration, velocity, and displacement. Real earthquakes, of course, have finite duration and a range of frequencies, and therefore as expected the ratios of spectral to ground-motion peaks are much less than. Nevertheless, it seems intrinsically more probable that the ground-motion peaks should be related to the spectral peaks than to spectral values at a fixed period. Indeed, 25 years ago, Newmark and Hall [982] proposed that design spectra should be constructed on the basis of constant ratios to peak ground-motion parameters and Eurocode 8 [EN998-, 24] also implies that these ratios can be taken as constant, as shown in Table. 2.2 Methodology Adopted in this Study When plotting the 5% damped spectrum of a typical ground-motion record, it is usual to find that peaks in spectral acceleration occur at a number of different periods, and the same applies to spectral velocity and displacement. For each spectral parameter, one peak will be greater than the rest, and so at first sight the problem is that of finding the value of this maximum peak, and hence its ratio to the corresponding peak ground-motion parameter. When dealing with the spectra of individual records, this approach works well, and yields stable relationships of relatively low variability. However, when dealing with smoothed code design spectra or those produced by ground-motion equations, this approach is not correct. Code spectra invariably have a single maximum value for spectral acceleration and velocity; in the writer s experience, the same is true for spectra generated from ground-motion equations. These maxima represent an average of the spectral values of the individual records used to generate the smoothed spectrum, but in general the true peaks of the spectra of individual records will occur at a range of different periods. Therefore, by relating PGV to the peak spectral velocity of each individual record, a true ensemble average value is not obtained, because the spectral values are always at their peak, rather than some being above the ensemble average and some below. The same applies to acceleration and displacement values. The corresponding ratios of peak spectral value to peak ground-motion value are higher than those relating to a smoothed spectrum. It would therefore be expected that finding the ratio of individual peak spectral values to peak

4 TABLE Ratios of peak 5% damped spectral values to peak motion parameters, from various sources This study Newmark & Hall [98] Malhotra [26] Individual peaks Peaks based on smoothed trend Mean Mean + σ Eurocode 8 Mean Mean + σ (Note ) Mean Mean + σ Mean Mean + σ Peak SA PGA Peak SV PGV (not defined) Peak SD PGD Note : ±σ confidence limits are not stated by Malhotra, but are drawn on his Figure 8, from which the values given in this table have been derived. 5

5 6 E. Booth ground-motion parameter, and applying this parameter to a smoothed spectrum to predict PGA, PGV, or PGD, would result in an underprediction of the ground-motion parameter. This study found that the ratios based on individual peaks gave results that were about 4% higher than those based on the smoothed trend of individual records, as shown in Table. The method adopted for finding the smoothed, trend spectra of individual records is described in the following paragraphs. The first attempt at finding the trend spectrum for an individual record was based on finding the period corresponding to the peak spectral value at high damping levels. The 5% damped spectral value was then found at this period, and the ratio to peak groundmotion parameter was calculated. It was hoped that by finding the peak period for increasing values of damping, an underlying trend would be established. This procedure did indeed result in a lowering of the spectral to ground-motion peak ratios as the damping level increased. However, it was hoped that the ratio would show convergence to a stable value at very high levels of damping. No such convergence was found for damping levels up to %. Therefore, another approach was used. Moving averages of the 5% damped spectra of individual records were calculated. The period ranges for the averaging, to either side of the period of interest, are shown in Table 2. Two different ranges were used for spectral velocity and displacement and three for acceleration, in order to see if the results were dependent on the period range adopted. For most records, the averaged spectra thus produced were still found to contain a number of peaks, often over a considerable period range. The period adopted as that corresponding to the spectral peak of the underlying trend was taken as the weighted average of the periods of spectral peaks within 75% of the overall maximum peak value. The weights for calculating the average period were the values of the spectral peaks (Eq. 2.). Here, SV Peak, is the first peak spectral velocity of at least 75% of the overall peak and T Peak, is the corresponding period. Similar expressions were used for spectral acceleration and displacement. T Peak, Trend ( T., SV, + T., SV, + T., SV = ( SV + SV + SV...) Peak Peak Peak 2 Peak 2 Peak 3 Peak, 3 Peak, Peak, 2 Peak, 3...) (2.) The ratio of spectral peak to ground-motion peak for each record was then based on the 5% damped spectral value at T Peak,Trend. The average ratio was calculated in this way for the averaging period ranges shown in Table 2; the average ratio was found vary within a range of less than ±5% for the averaging periods investigated for velocity and acceleration, and by less than ±% for the averaging periods investigated for displacement. The methodology described above was then applied to PGV ratios for 6 horizontal records from 55 mainly North American earthquakes, with distances from the causative fault reported as ranging from under km to 223 km. The records were chosen from a larger data set of 292 horizontal records originally assembled by Abrahamson and Silva TABLE 2 Period ranges for calculating moving average values of 5% damped spectra Period ranges (seconds) SA, spectral acceleration ±.25 ±.5 ±.75 SV, spectral velocity ±.625 ±.25 SD, spectral displacement ±.25 ±.5

6 Estimating Peak Motions from Spectral Ordinates 7 [997]; 3 records were excluded, either because the PGV reported in the database did not match that calculated by single integration of the acceleration time histories to within 5%, or because the calculated period at peak spectral velocity exceeded 2.4 sec; the latter value was chosen as 8% of 3 sec, the maximum period to which spectral values were calculated. The same dataset was used for calculating peak spectral displacement to PGD ratios; the results were based on,58 of the records, excluding those with a period at peak spectral displacement exceeding 2 sec, since spectral values were only calculated to a maximum period of 5 sec. Also, records were rejected where the reported PGD in the database did not match that calculated by double integration of the time histories to within 5%. Integration of acceleration time histories to produce velocities and (especially) displacements is sensitive to the processing applied to the accelerograms; the only check carried out for this study was, as reported above, that peak velocity and displacement agreed within 5% (for PGV) and 5% (for PGD) of the values reported by Abrahamson and Silva [997]. The results are shown in Figures and 2 and summarized in Eqs. (2.2) and (2.3). PGV = (peak 5% damped spectral velocity from smoothed spectrum) / 23. β ( coefficient of variation) = 6. (2.2) PGD = (peak 5% damped spectral displacement from smoothed spectrum)/ 23. β ( coefficient of variation) = 5. (2.3) The same methodology was then applied to finding the ratio between peak spectral acceleration and peak ground acceleration, PGA. Of course, PGA is known with certainty from the response spectrum, so the ratio is not useful for predictive purposes but is still of intrinsic interest. Using exactly the same database of,6 records used to derive Eq. (2.2), the following relationship was found. PGΛ = (peak 5% damped spectral acceleration from smoothed spectrum) / 265. β ( coefficient of variation) = 3. (2.4) 3. Prediction of Peak Ground-motion Parameters from Random Vibration Theory 3. Theoretical Basis for Prediction of PGV The peak spectral to ground-motion ratios calculated as described above were derived purely empirically. They were also to some extent influenced by the method used to find the peaks in the spectral records. An attempt was therefore made to develop a method of calculation based on a firmer theoretical basis. It used the fact that there is a reasonably close relationship between the pseudo-spectral velocity spectrum and the Fourier spectrum of the stationary part of the underlying record. Together with knowledge of the record duration and predominant period, this enables statistical estimates to be made of PGV and PGD; the details are now explained. The Fourier spectrum of the stationary part of a record can be shown to be related to the % damped pseudo-velocity response spectrum, but usually only a 5% spectrum will be available. Therefore, an adjustment to convert from 5% to % damping must be made.

7 8 E. Booth R 2 =.6 Predicted PGV Observed PGV Predicted PGV Observed PGV.. PGV (cm/sec) Period at peak SV a) Peak SV/2.3: eq.(2.2) R 2 =.5 Predicted PGV Observed PGV Predicted PGV Observed PGV.. PGV (cm/sec) Period at peak SV b) Random vibration theory: eq.(3.) R 2 =.34 Predicted PGV Observed PGV Predicted PGV Observed PGV.. PGV (cm/sec) Period at peak SV c) SA(.5)/2: Bommer and Alarcon [26] FIGURE Predictions of PGV from spectral ordinates by three methods. Peak SD/2.3: eq. (2.3) Random vibration theory: eq. (3.3) Predicted PGD Observed PGD Predicted PGD Observed PGD... PGD (cm)... PGD (cm) FIGURE 2 Predictions of PGD from spectral ordinates by two methods. Denoting the 5% damped acceleration spectrum as SA 5% (T) and the % damped velocity spectrum as SV % (T) (where T is the structural period), one can write: SV ( T ) = ( T / 2p ). f ( T ). SA ( T ) (3.) % 5%

8 Estimating Peak Motions from Spectral Ordinates 9 where f(t) is some function of structural period, which will depend on the ground-motion record in question. At zero period, f() is exactly, and then rises to a peak at the predominant periods of the ground-motion. It then tends to reduce at longer periods, but will never fall below. Eurocode 8 Part [EN998-, 24] eq. (3.6) gives a period-independent value for damping correction, understood to have been derived for the range 5% to 28% damping (although the lower limit is not stated). Assuming the equation to be valid at % damping gives a period independent value for f: /2 f = {/(5 + )} =.4 (3.2) Having obtained the % damped velocity spectrum from Eq. (3.), the Fourier acceleration spectrum can be estimated. Hudson [962] has shown that the former represents a fairly close upper bound to the latter. Writing F a as the Fourier acceleration spectrum, the relationship may be expressed as: Fa = g( T). SV% ( T) (3.3) where g(t) is some dimensionless parameter related to period, and depending on the record in question, but never greater than one. The Fourier acceleration spectrum can then be converted to the Fourier velocity spectrum by dividing by the circular frequency (Eq. 3.4), and the rms (root mean square) velocity, v rms, follows directly from the well known relationship (Parsifal s equation), given here as Eq. (3.5). Fv = ( T / 2p ). Fa (3.4) v = / p { F } dt ] = [ / p { T/( 2p). F } dt rms T = T = 2 / 2 2 / 2 v a [ ] (3.5) The rms velocity can then be used to obtain a statistical estimate of PGV. Boore [23] reports that the following relationship applies 2 / 2 / PGV = v [{ 2 ln( N )} /{ 2 ln( N )} ] (3.6) rms v v Here, N v is the number of zero crossings during the stationary part of the velocity record; in other words, the number of times the velocity changes sign. This will depend on the predominant periods of the record, and the duration of the strong motion portion of the record. In fact, the right hand side of Eq. (3.6) varies only slowly with N v, and so deriving PGV from v rms requires only a crude estimate of N v. Calculating PGV from a 5% damped response spectrum using Eqs. (3.), (3.3), (3.4), (3.5), and (3.6) therefore requires knowledge of the following:

9 2 E. Booth. f (T), the ratio of % to 5% damped spectral ordinates 2. g(t), the ratio of % damped velocity response spectrum to Fourier acceleration spectrum 3. N v, the number of zero crossings of the record, or equivalently its duration and predominant period. Initially, f (T) and g (T) were taken as period independent constants of.4 and.8, respectively. Reasonable assumptions were made on predominant period and duration, based on the reported soil type at the location of the recording and the magnitude of the causative earthquake. These values produced good predictions for PGV. However, it was realized that the form of the equations can also be used to predict PGA, which is of course known with certainty from the 5% damped response spectrum as the zero period acceleration. It can easily be shown that Eq. (3.7) applies. T = [ / p { Fv } dt] PGV = PGA T = 2 [ / p { F } dt] a 2 / 2 2 / / 2 / 2. {(ln 2 Nv ) /(ln Nv ) } 2 / 2 / { 2(ln N ) /(ln N ) } a a (3.7) where N a is the number of zero crossings of the acceleration record. In general, N a will be greater than N v. Writing { 2(ln Nv) +. 58/(ln Nv) } X = (3.8) 2 / 2 / { 2(ln N ) /(ln N ) } a 2 / 2 / X is relatively insensitive to variations in N a and N v, and would be expected to be in the range.8 to.9 for most records. Combining eqs. (3.7) and (3.8) then gives a PGV = X. PGA T = [ ] / p { f ( T ). g( T ). SA ( T ). ( T/ 2p) } dt [ T = 5% 2 2 / 2 / p { f ( T). g( T). SA ( T). ( T/ 2p)} dt] 5% 2 / 2 3 (3.9) Making the approximation that f (T) and g(t) are period independent greatly simplifies Eq. (3.9) to yield Eq. (3.) T = [ { SA ( T ). 5% ( T / 2p) } dt] PGV = X. PGA T = [ { SA ( T ). ( T/ 2p)} dt] / 5% 2 2 / (3.) Given a 5% damped response spectrum, the only unknown on the right-hand side of Eq. (3.) is the parameter X, which as noted above might be expected to be around.85. In fact, analysis of the same,6 records described in Sec. 2.2 gave a best fit value of X as.79, assuming a log normal distribution of the ratio of predicted to actual PGV. X allows not only for the differing values of N a and N v but also the bias introduced by taking f(t) and g(t) as frequency independent, and (more fundamentally) the differences

10 Estimating Peak Motions from Spectral Ordinates 2 between real records and the idealized records assumed by the theory. Also, the integration cannot in practice extend over an infinite period range. Nevertheless, the fact that X falls within the range expected and the good agreement between predicted and actual PGV shown in Figure gives some confidence that Eq. (3.) should hold for a wider range of records than those investigated in this study. For this study, integration was carried out numerically with a constant period increment ΔT of.25 sec. Integration was carried out to upper limits of both 3 sec and 5 sec with little improvement in results for the larger period range, although it should be noted that the few records with periods at peak spectral velocity exceeding 2.4 sec were excluded from the dataset. Also, as reported later, integration to 5 sec did yield significant improvement in results when applied to near-fault records. Rounding up the value of X gives the final recommended form as Eq. (3.). Note that it is assumed that consistent units are used: e.g., PGV in cm/sec, PGA in cm/sec 2 and T in seconds. [ { SA5 ( T ). % ( T / 2p) } dt] PGV = 8. PGA T = 3 [ { SA ( T ). ( T / 2p)} dt] / β (coefficient of variation) =. T = 3 5% 2 2 / (3.) 3.. Extension of random vibration theory to prediction of PGD. Eq. (3.) can easily be extended to PGD: T = / [ { SA5% ( T ).( T / 2p) } dt] PGV = X. PGA T = [ { SA ( T ).( T / 2p)} dt] / 5% (3.2) Here, X allows for the much smaller number of zero crossings in the displacement trace, compared with acceleration trace. However, the theoretical basis assumes that the displacement record over the strong motion part of the record has the form of a stationary random vibration, which is far from the case. Nevertheless, it was decided to see how reliable Eq. (3.2) was in predicting PGD for the same,58 records used to investigate peak spectral displacement to PGD ratio. Somewhat to the author s surprise, using an integration range up to 5 sec and a period increment ΔT of.25 sec, Eq. (3.3) provided a slightly better fit to the results than that based on peak spectral displacement. However, the very long spectral periods involved in the calculation mean that the results are not very reliable, as indeed is the case for the underlying theory when applied to displacements. T = / [ { SA5% ( T ).( T / 2p) } dt] PGV = 7. PGA T = [ { SA ( T ).( T / 2p)} dt] / 5% (3.3) β (coefficient of variation) =.2 Results are shown in Figure 2.

11 22 E. Booth 4. Prediction of Peak Ground-motion Parameters from.5 Second Spectral Acceleration Bommer and Alarcon [26] have proposed that PGV implied by a smoothed spectrum can be predicted from Eq. (4.). PGV = SA5% ( T = 5. )/ 2 (4.) where SA 5% (T =.5) is the 5% damped pseudo spectral acceleration at.5 sec. PGV and SA 5% (T =.5) must be in consistent units (eg cm/sec and cm/sec 2 ). For the data set used in this study, the coefficient of variation β was found to be.9, and the average of predicted/actual PGV was.95. The results are plotted on Figure. It is easy to show that Eq.(4.) is equivalent to PGV = SV5%( T = 5. )/ 59. (4.2) Assuming that Eq. (2.2) is reliable, and comparing Eq. (4.2) with Eq. (2.2), would lead to the expectation that Eq. (4.) would tend to overpredict PGV when the period at peak spectral velocity was around.5 sec, and underpredict it when the period was much greater or much less. The trend lines plotted on Figure support this expectation; the correlation coefficient R between SA 5% (T =.5)/2.3 and predominant period for the trendline is shown on Figure as 34%. For the other two methods, the trendlines are practically constant at and the correlation coefficients are less than 2%, suggesting a very low correlation with period. Equation (4.) has the advantage that it is much easier to calculate than Eqs. (2.2) or (3.), because only one spectral period is involved. By contrast, to solve Eq. (2.2), the peak spectral velocity must be found, involving calculation at a range of periods, and Eq. (3.) involves a complex integration over many periods. 5. Malhotra s Smoothed Spectrum Malhotra [26] has proposed a method for constructing a smooth design spectrum from the peak ground parameters PGA, PGV, and PGD. The method is an extension of that proposed by Newmark and Hall [98]. It involves normalising the period scale by dividing all periods by T cg, given by pgd Tcg = 2p. (5.) pga The spectral ordinates are normalized by the factor pga. pgd. Malhotra used a database of 63 horizontal records to derive a mean trend through the 5% damped spectra of all 63 records. The results of particular relevance to this study are the peak spectral to peak ground-motion parameters, which are shown in Table. It can be seen that the ratios are significantly lower than found in this study, and (for acceleration and velocity) very close to the values given by Newmark and Hall [98]. Figure 3 compares Malhotra s central period T cg with T peak,trend, the period corresponding to peak spectral velocity from this study, taking the trend of each individual

12 Estimating Peak Motions from Spectral Ordinates 23 Malhotra s Tcg Period at peak SV Period at peak SV (secs) 3 FIGURE 3 Comparison of T cg : Eq. (5.) from Malhotra [26] with period at peak spectral velocity: Eq. (2.) from this study. spectrum. The comparison is based on the,58 records from the Abrahamson and Silva [997] database where the reported and calculated PGD differed by less than 5%. It can be seen that T cg predicts the average value of the period at peak spectral velocity rather well over a wide range of periods; Malhotra give a value of. T cg compared with a logarithmic average of.9t cg for the data of Figure 3. However, there is a wide scatter. Since Malhotra s method involves smoothing over the combined results, while this study bases the spectral ratio on the trend of each individual spectra, it would be expected that Malhotra s results would result in a much greater relative reduction in the peak spectral ordinates. As seen from Table, Malhotra s spectral ratios are indeed considerably lower than those from this study. Which method of smoothing is more appropriate is discussed later in this article. 6. Validation of Results Two validation tests were carried out. The first involved comparison of the predictions of peak ground-motion parameters from the methods discussed in this paper with those from ground-motion predictive equations. It was intended as a general check on the methods developed, and in particular to test whether the smoothing methods used to derive Eq. (2.2) were appropriate. The second test involved using the methods of this paper on a set of 8 ground-motion records taken near the source of large earthquakes. It was intended to test the validity of the predictions in the presence of near-fault effects, namely directivity and velocity flings. 6. Comparison with the Values given by Ground-motion Predictive Equations Six ground-motion predictive equations were chosen which gave both 5% damped spectral ordinates and peak ground velocity, based on a consistent dataset. The sources used were Tromans and Bommer [22] together with Ambraseys et al. [996] which used

13 24 E. Booth essentially the same dataset, Campbell [997], Sadigh [997, 998], Molas and Yamazaki [995, 996], Joyner and Boore [988], and Sabetta and Pugliese [996]. A range of magnitudes between 5 and 8, of distances between km and km and soil types were taken, to give a representative spread of results. The magnitudes and distances were chosen for each predictive equation to be within the ranged of applicability stated by the authors. The results are shown in Figure 4 and the averages and dispersions are shown in Table 3. It can be seen that Eq. 2.2 (peak SV/2.3) tends to underestimate results by about %, while Malhotra s peak SV/.65 overestimates by about 25%. Bommer and Alarcon s SA(.5)/2 (Eq. 4.) gives an average ratio close to but has the largest scatter. Random vibration theory overestimates PGV on average by 4%, but the results for Campbell s hard rock class show a consistently high ratio of between.7 and 2.3. Removing the hard rock result reduces the overestimate to 7% and gives the lowest scatter of the four relationships examined. Campbell [private communication] advises that hard rock refers to very Tromans & Bommer Sadigh et al Campbell - hard rock Campbell - soft rock & soil Molas and Yamazaki Joyner & Boore Sabietta & Pugliese Tromans & Bommer Sadigh et al Campbell - hard rock Campbell - soft rock & soil Molas and Yamazaki Joyner & Boore Sabietta & Pugliese eq predicted pgv from grnd motion predicted pgv from grnd motion pgv pgv eq pgv (cm/sec) pgv (cm/sec) a) Peak SV/2.3: eq. (2.2) b) Random vibration theory: eq. (3.) Tromans & Bommer Sadigh et al Campbell - hard rock Campbell - soft rock & soil Molas and Yamazaki Joyner & Boore Sabietta & Pugliese Tromans & Bommer Sadigh et al Campbell - hard rock Campbell - soft rock & soil Molas and Yamazaki Joyner & Boore Sabietta & Pugliese eq predicted pgv from grnd motion predicted pgv from grnd motion pgv eq pgv... pgv (cm/sec) c) SA(.5)/2: Bommer and Alarcon [26]... pgv (cm/sec) d) Peak SV/.63: Malhotra [26] FIGURE 4 Comparison of predictions of PGV from spectra generated by ground motion equations with direct prediction from a ground motion equation.

14 Estimating Peak Motions from Spectral Ordinates 25 TABLE 3 Summary of results shown in Figure 4 averaged over six ground motion predictive equations Peak SV/2.3: eq (2.2) Random vibration theory: eq (3.) SA(.5)/2: Bommer & Alarcon [26] Peak SV/.63: Malhotra [26] Logarithmic average of (predicted PGV) (PGV from predictive equation) Coefficient of variation β 84 percentile confidence range Note Note 2 Note Note 2 Note Note to.7.7 to to.5.87 to to.38.7 to to to.62 Note : Figures including Campbell [997] for hard rock. Note 2: Figures excluding Campbell [997] for hard rock. hard basement rock material; the classification of rock in most ground-motion equations is considerably softer. It is not clear why the discrepancy between random vibration theory and Campbell s hard rock results should occur; the fact that it does not apply to the other methods suggests that it is associated with the long period spectral response included in Eq. (3.) but not the other methods. This is an issue which merits further study. 6.2 Validity for Near-fault Records PGV and PGD were estimated for 8 near-fault records shown in Table 4. They are taken from Tables and 2 given in Bray and Rodriguez-Marek [24] which were chosen to investigate one type of near-fault effect namely forward-directivity. Bray and Rodriguez-Marek studied recordings from 54 stations which captured near-fault groundmotions from 3 earthquakes of magnitude range 6. to 7.. All but two of the sets of recordings can be downloaded from the Pacific Earthquake Engineering Research (PEER) strong motion database ( Two near-fault Chi-Chi records were substituted for two Chi-Chi records attributed to CDMG and used by Bray and Rodriguez-Marek, since they do not appear to be in the PEER database. The two horizontal components of each record (shown in Table 4 as x and y components) are taken separately, making a total 8 records. The database contains records which exhibit fling as well as forward-directivity effects, but (unlike the work of Bray and Rodriguez-Marek [24], which was for a different purpose) no attempt was made to remove the former effect. Because of the long periods involved, the maximum spectral period investigated in the analysis for peak SV/2.3, Eq. (2.2), was increased from 3 sec to 5 sec; similarly, for random vibration theory, the upper integration limit in Eq. (3.) was increased to 5 sec. Results are shown in Table 5 and Figure 5. It can be seen that PGV is best predicted by random vibration theory, with an average and coefficient of variation β almost the same as was found for the full Abrahamson & Silva dataset. Peak SV/2.3 predicts less well, with an average value of predicted to observed PGV of.8. However, the β value is

15 TABLE 4 Near-fault records used for Figure 5 PGA (g) (Note 2) PGV (cm/s) (Note 2) PGD (cm) (Note 2) Earthquake M (Note ) Year Station R (km) x y x y x y Parkfield Cholame # Temblor San Fernando Pacoima dam Imperial Valley Brawley airport EC County center FF EC Meloland overpass FF El Centro array # El Centro array # El Centro array # El Centro array # El Centro array # El Centro array # El Centro array # El Centro Diff. Array Holtville Post Office Westmorland Fire Sta Morgan Hill Coyote Lake Dam (SW Abut) Gilroy Array # Superstition Hills(B) El Centro Imp. Co. Cent Parachute Test Site Loma Prieta Gilroy Gavilan Coll Gilroy Hist. Bldg Gilroy Array # Gilroy Array # Gilroy Array # LGPC

16 Saratoga Aloha Ave Saratoga W Valley Coll Erzincan Erzincan Landers Lucerne Northridge Jensen Filter Plant LA Dam Newhall Fire Sta W. Pico Canyon Rd Pacoima Dam (downstr) Pacoima Dam (upper left) Rinaldi Receiving Sta Sylmar Converter Sta Sylmar Converter Sta East Olive View Med FF Kobe KJMA Takarazuka OSAJ Takatori Kocaeli Arcelik Duzce Gebze Chi-Chi TCU TCU TCU TCU TCU TCU Duzce Bolu Note : Magnitudes given are moment magnitudes. Note 2: x and y refer to the two horizontal components of motion given in the PEER database ( 27

17 28 E. Booth TABLE 5 Summary of results shown in Figure 5 for near-fault records of Table 4 Logarithmic average of predicted PGV actual PGV Coefficient of variation β 84 percentile confidence range Peak SV/2.3: eq (2.2) to.6 Random vibration to.3 theory: eq (3.) SA(.5)/2: Bommer & Alarcon [26] to.7 unchanged at.6. For SA(.5)/2, eq. (4.), β increases from.9 for the Abrahamson & Silva dataset to.23 for the near-fault records, and the average drops from.95 to.63, confirming Bommer and Alarcon s [26] observation that the relationship is not advisable for use where near-fault effects are present. Trendlines have been fitted to Figure 5, and the correlation coefficients R 2 are shown on the graphs. It can be seen that peak SV/2.3 and random vibration theory show little or no dependence on T peak,trend, the period at peak spectral velocity. However, SA(.5)/2 shows a significant dependence on T peak,trend, with same marked tendency shown in Figure c to underpredict PGV for large values of T peak,trend and to overpredict at a period of.5 sec. 7. Discussion The method used in this study to derive spectral ratios is based on the characteristics of individual records. It would be expected to produce the most representative spectral shapes corresponding to real ground-motions. A more drastic degree of smoothing is likely to apply in the derivation of ground-motion equations, because of the wide range of actual soil types and site periods lumped together under one soil type. Thus, the spectral value given by a ground-motion equation for a particular period, magnitude, epicentral distance, and soil type average will be based on the average of spectral values at different positions relative to their spectral peaks, which will occur at periods that are significantly affected by the site period and hence also by the soil conditions. This range of periods corresponding to spectral peak will tend to lower spectral values at peaks but increase them elsewhere. Therefore, the values in this study would be expected to underestimate the PGV corresponding to a given peak spectral velocity obtained from a ground-motion equation. Table 3 suggests that this is indeed the case, the underestimate being around %. For similar reasons, the smoothing method used by Malhotra [26] also tends to lower the spectral peak, and Table 3 suggests that this effect may be as much as 35% for PGV. The random vibration theory considers values over the entire spectral range, and so the effect of lower spectral peak from a ground-motion equation might be expected to be offset (at any rate to some extent) by the higher values away from the peak. The results in Table 3 excluding Campbell s hard rock tends to support this expectation, but inclusion of the Campbell hard rock results makes the argument rather less sustainable. These are issues here which deserve further investigation. Deriving PGA, PGV, and PGD from smoothed response spectrum may be useful for finding real time histories which match a specified design spectrum, by choosing records which have similar values of the three peak ground-motion parameters. This may be attractive because it might ensure that matching the three parameters would help a similar

18 Estimating Peak Motions from Spectral Ordinates 29 R 2 =.9 Predicted PGV Observed PGV Period at peak SV (seconds) a) Peak SV/2.3: eq (2.2) R 2 =.3 Predicted PGV Observed PGV Period at peak SV (seconds) b) Random vibration theory: eq (3.) 5 R 2 =.5 Predicted PGV Observed PGV Period at peak SV (seconds) c) SA(.5)/2: Bommer & Alarcon [26] FIGURE 5 Predictions of PGV from spectral ordinates by three methods for near-fault events.

19 3 E. Booth matching of underlying frequency content over a broad range of periods. A trial use of this application by the author proved promising, but it was not rigorously tested, and would merit further investigation. It is worth considering what degree and method of smoothing is appropriate for a given circumstance. The method used in this paper in deriving Eqs. (2.2) to (2.4) might be expected to produce the most representative spectral shapes for specific seismological and geotechnical conditions. It is probably best when specifying the PGA, PGV, and PGD values of individual records required to be compatible with a smoothed design spectrum. However, for structural design purposes, the uncertainties involved in all of the seismological and geotechnical parameters point to a broader envelope spectrum being required, perhaps as provided by Malhotra at the mean plus a suitable number of standard deviations. Once again, these are issues which deserve further investigation. 8. Conclusions. The following ratios of peak spectral values to peak ground-motion parameters were found from the smoothed spectra of over, horizontal strong groundmotion records. PGA = peak SA PGV = peak SV ( smoothed) ( smoothed) / / PGD = peak SD ( smoothed) / 23. where peak SA (smoothed), peak SV (smoothed), and peak SD (smoothed) are mean values of the 5% damped peak spectral acceleration, peak spectral velocity and peak spectral displacement, taken from a smoothed response spectrum. 2. The coefficient of variation β for these relationships is in the range.3 to The relationships appear to be valid over a wide range of seismological conditions, although they become somewhat less reliable in the presence of near-fault effects. They are not biased with respect to magnitude, distance or soil type. 4. These ratios of spectral ordinate to corresponding peak ground-motion parameter may be about % too high when applied to the smoothed spectra derived from ground-motion equations, at any rate for PGV. 5. The ratios are significantly higher than those originally reported by Newmark and Hall [98] and more recently by Malhotra [26]. The Newmark and Hall and Malhotra ratios appear to be around 25% too low, when used to estimate PGV from smoothed spectra derived from ground-motion predictive equations. 6. The ratios of spectral peaks to PGA and PGD found in this study are close to those adopted in Eurocode A more reliable estimate of PGV (Eq. 3.), based on random vibration theory and involving integration of the (smoothed or unsmoothed) 5% damped response spectrum, reduces the β value to.9. Equation (3.) works well in the presence of near-fault effects, although it is advisable to increase the upper limit of integration in Eq. (3.) from 3 sec to 5 sec. 8. The same random vibration theory can be extended to provide an estimate of PGD (Eq. 3.3), but it does not result in a significantly improved estimate, compared with that based on peak spectral displacement (Eq. 2.3).

20 Estimating Peak Motions from Spectral Ordinates 3 9. The random vibration theory is therefore the most reliable of the methods studied here, but it is also the most complex. However, it is still easily implementable in a spreadsheet. One of its advantages with respect to use with smoothed spectra is that it should be largely independent of the smoothing method used in the source spectrum.. When applied to individual records, the estimate of PGV based on 5% spectral ordinate at.5 sec period gives less accurate results than those based on either peak SV (smoothed) /2.3 or random vibration theory, and appears consistently to underestimate PGV for records with unusually long or short predominant periods. In particular, it is not reliable in the presence of near-fault effects. However, when compared with results from predictive equations for PGV, this method gives results broadly comparable to the other two. Moreover, the method is much the simplest to implement.. Deriving PGV from smoothed response spectra may be useful where damage estimates are required corresponding to spectra derived from ground-motion predictive equations providing spectral ordinates but not peak ground velocity. PGV, as discussed by Bommer and Alarcon [26], appears to be one of the best single predictors of structural damage and geotechnical effects. 2. Deriving PGA, PGV, and PGD from smoothed response spectra may also be useful for finding real time histories which match a specified design spectrum, by choosing records which have similar values of the three peak ground-motion parameters. Acknowledgments The strong motion database used in this study was kindly provided by Walt Silva of Pacific Engineering and Analysis. Extensive and valuable discussions on the study and its interpretation were held with Julian Bommer and John Alarcon of Imperial College, Praveen Malhotra of FM Global and Kenneth Campbell of EQECAT. An anonymous reviewer sent extensive and very useful comments and suggestions. The support and encouragement of all these people is gratefully acknowledged. References Abrahamson, N. A. and Silva, W. J. [997] Empirical Response Spectra Attenuation Relations for Shallow Crustal Earthquakes, Seismological Research Letters, 68(), Ambraseys, N. N, Simpson, K. A. and Bommer, J. J. [996] The prediction of horizontal response spectra in Europe, Earthquake Engineering and Structural Dynamics, 25, Bommer, J. J. and Alarcon, J. E. [26] The prediction and use of peak ground velocity, Journal of Earthquake Engineering, (), 3. Bray, J. D. and Rodriguez-Marek, A. [24] Characterization of forward-directivity ground motions in the near-fault region, Soil Dynamics and Earthquake Engineering, 24(2), Boore, D. M. [23] Simulation of ground motion using the stochastic method. Pure and Applied Geophysics, 6, Campbell, K. W. [997] Empirical near-source attenuation relationships for horizontal and vertical components of peak ground acceleration, peak ground velocity and pseudo-spectral acceleration response spectra, Seismological Research Letters, 68(), EN998- [24] Eurocode 8: Design of structures for earthquake resistance. Part : General rules, seismic actions and rules for buildings, (CEN, Brussels). Hudson, D. E. [962] Some problems in the application of spectrum techniques to strong motion analysis. Bulletin of the Seismological Society of America, 53(2),

21 32 E. Booth Joyner, W. B. and Boore, D. M. [988] Measurement, characterization and prediction of strong ground motion, Proceedings of Earthquake Engineering & Soil Dynamics II, Park City, Utah, Malhotra, P. K. [26] Smooth spectra of horizontal and vertical ground-motions, Bulletin of the Seismological Society of America, 96(2), April. Molas, G. L. and Yamazaki, F. [995] Attenuation of earthquake ground-motion in Japan including deep focus events, Bulletin of the Seismological Society of America, 85(5), Molas, G. L. and Yamazaki, F. [996] Attenuation of response spectra in Japan using new JMA records, Bull. Earthquake Resistant Struct., 29, Newmark, N. N. and Hall, W. J. [98] Earthquake Spectra and Design. Earthquake Engineering Research Institute, Berkeley CA. Sabetta, F. and Pugliese, A. [996] Estimation of response spectra and simulation of nonstationary earthquake ground motions, Bulletin of the Seismological Society of America, 86(2), Sadigh, R. K., Chang C. Y., Egan, J. A., Makdisi, F., and Youngs, R. R. [997] Attenuation relationships for shallow crustal earthquakes based on Californian strong motion data, Seismological Research Letters, 68(), Sadigh, R. K. and Egan, J. A. [998] Updated relationships for horizontal peak ground velocity and peak ground displacements for shallow crustal earthquakes, Proc. of the Sixth U.S. National Conference on Earthquake Engineering, Seattle, Paper 37. Tromans, I. J. and Bommer, J. J. [22] The attenuation of strong-motion peaks in Europe, Proc 2 th European Conference on Earthquake Engineering, London.