Seismological Research Letters
 © 2007 by the Seismological Society of America
Abstract
Performancebased earthquake engineering (PBEE) concepts dictate that the return period of the seismic design ground motions be known. While the NEHRP (2001) seismic design spectrum is constructed using two spectral accelerations that have a 2,475year return period, the overall spectral shape is not representative of a uniform hazard spectrum. Consequently, other than the two spectral accelerations used to construct the NEHRP spectrum, the return periods of the spectral accelerations defined by the NEHRP spectrum are unknown and may vary significantly. An alternative spectrum is proposed for firm rock sites. The proposed parabolic spectrum is constructed using three spectral accelerations (i.e., pga, S_{S}, and S_{1}) and a calibration parameter T_{pga}. Parabolic spectra were computed for approximately 500 cities across the conterminous 48 states using the pga, S_{S}, and S_{1} having 2,475year return period, as determined by the U.S. Geological Survey National Seismic Hazard Mapping Project. The correlation coefficients (r ^{2}) of the parabolic spectra to spectral accelerations for several oscillator periods having uniform probabilities of exceedance were computed for the cities. The results confirm that the spectral accelerations defined by the parabolic spectra have approximately constant return periods.
INTRODUCTION
Over the past several decades, significant advances have been made in earthquake engineering and related sciences. However, during this same period, the seismic risk to the world's infrastructure has increased, as quantified in terms of losses resulting from earthquakes (figure 1). To reverse this trend, the earthquake engineering community has been moving toward performancebased earthquake engineering (PBEE), or more broadly, performancebased design (PBD).
The adaptation of building codes in the United States and around the world to embrace PBEE concepts represents a paradigm shift in the engineering profession's approach to seismic risk mitigation. As opposed to the traditional design approach wherein life safety was the primary design goal, structural design in PBEE is governed by a targeted performance level that has an associated annual probability of exceedance. Proper implementation of PBEE requires knowledge of both the fragility of the structural system and the probabilistic seismic hazard. In relation to this latter requirement, the ground motions used to construct the seismic design spectrum per the 2000 edition of the National Earthquake Hazards Reduction Program (NEHRP) Recommended Provisions for Seismic Regulations for New Buildings and Other Structures (NEHRP 2001) were determined using formal probabilistic concepts (e.g., Frankel et al. 2000).
The NEHRP (2001) seismic design spectra are constructed from the 5% damped 0.2 and 1.0sec spectral accelerations having an approximately 2,475year return period (or correspondingly, a 2% probability of exceedance in 50 years: 2% PE 50yrs). However, the general shape of the NEHRP spectrum is based on the Newmark and Hall (1982) design spectrum. Consequently, the design spectral accelerations across all periods do not necessarily have the same return period, which is inconsistent with PBEE concepts. The focus of the study presented herein is the selection of a design spectral shape for rock motions (i.e., NEHRP site class B) that better represents a constant return period across all frequencies than the NEHRP (2001) design spectrum, and that has a simple enough functional form to be readily adopted by building codes.
This paper first presents a brief overview of the framework of PBEE, then reviews the NEHRP (2001) design spectrum shape, then follows with a presentation and discussion of an alternative spectral shape (henceforth referred to as the parabolic spectrum). Next, we compare the NEHRP and parabolic spectral shapes using spectra for four cities to illustrate trends. We further explore these trends using data from approximately 500 cities that are geographically dispersed across the conterminous 48 states. Finally, we present and discuss a contour map for the calibration parameter (T_{pga}) for the parabolic spectrum for the conterminous 48 states in order to facilitate adoption of the parabolic spectrum by building codes. Factors for adjusting the firm rock parabolic spectrum to other NEHRP site classes and to damping levels other than 5% of critical are beyond the scope of the present paper and will be presented and discussed in subsequent papers.
THE FRAMEWORK OF PBEE
The ultimate goal of PBEE is to design a structure that has targeted annual probabilities of exceedance for one or more performance objectives (e.g., no damage to near collapse) (e.g., Fajfar and Krawinkler 1997; Mahoney and Hanson 1998; Priestley 2000; Chandler and Lam 2001; Bertero and Bertero 2002; Deierlein et al. 2003; Kramer et al. 2006). The Pacific Earthquake Engineering Research (PEER) Center's framework for PBEE consists of four steps: hazard assessment; structural/nonstructural component analysis; damage evaluation; and loss analysis or risk assessment, with subsequent steps conditionally linked to the previous steps. Consequently, hazard assessment, or more specifically, the quantification of the annual probability of exceedance of design ground motions, is integral to implementation of PBEE. While the implementation of PBEE in its purest form may never be realized in general practice, many of the PBEE concepts are inherent in recent building codes (e.g., NEHRP 2001). Figure 2 illustrates the current state of implementation of PBEE, per NEHRP (2001). In this figure, building performance objectives for three building categories (i.e., Group I, Group II, and Group III buildings) are specified as functions of ground motion level. The engineer only designs to one ground motion level (i.e., two thirds of the motions for the “maximum considered earthquake”), and the building performance objectives at the other ground motion levels are inherently achieved through structural detailing, etc., prescribed by the design code. The functional relationship among building performance objectives, ground motion levels, and the structural design specifications is largely based on experience derived from postearthquake investigations. However, efforts are underway (i.e., ATC 58, 2006) to establish a more rigorous functional relationship among these three components (personal communication between R. Green and Robert Hanson, 2006).
NEHRP (2001) DESIGN SPECTRUM
The NEHRP (2001) spectrum is constructed from 5% damped, 0.2 and 1.0sec spectral accelerations having a 2,475year return period. These spectral accelerations are designated as S_{S} and S_{1}, respectively. The NEHRP (2001) code is accompanied by a set of seismic hazard maps from which S_{S} and S_{1} can be determined for NEHRP site class B (i.e., firm rock) for any site within the United States. Color versions of the S_{S} and S_{1} contour maps obtained from the U.S. Geological Survey (USGS) Web site are presented in figures 3(B) and (C), respectively. (Note: Figure 3A is a color contour map for peak ground acceleration (pga) having 2,475year return period; this figure will be discussed later in the paper in relation to the proposed parabolic spectrum. Also, the maps in figure 3 do not include the deterministic caps inherent to the NEHRP maps.) Site response factors (F_{a} and F_{v}) are then applied to S_{S} and S_{1}, respectively, to adjust the spectral accelerations to site classes other than site class B (equation 1). However, because the focus of this paper is only on firm rock spectra, F_{a} and F_{v} are set equal to 1 herein. (1a) (1b) The design spectral values (i.e., S_{DS} and S_{D}_{1}) are then set equal to two thirds of the S_{MS} and S_{M}_{1} (equation 2) to provide a uniform margin against collapse of structures, i.e.: (2a) (2b) The NEHRP design spectrum, S_{a}(T), is then constructed from equation 3: (3) where and .
Figure 4(A) shows an example of the NEHRP design spectrum computed using equation (3). For comparison purposes, the NewmarkHall design spectrum is shown in figure 4(B) (Newmark and Hall 1982). The similarities in the two spectral shapes are readily apparent. Except for long periods where the NewmarkHall spectrum has a constant displacement region and the NEHRP spectrum does not, the two spectral shapes are almost identical. (Note: The design spectrum specified in NEHRP (2003) includes the constant displacement region at long periods, thus making it even more similar in shape to the NewmarkHall spectrum.) The similarity in the shapes of the NEHRP and NewmarkHall spectra is testament that the spectral amplitudes defined by the former are statistically correlated across frequencies, similar to actual ground motions. On the contrary, a uniform hazard spectrum (UHS) determined from a probabilistic seismic hazard analysis (PSHA) defines spectral amplitudes that are statistically independent across frequencies and that have a constant probability of exceedance (or return period). Consequently, a UHS for a given region does not necessarily resemble response spectra computed from ground motions recorded in that region. In the case of the NEHRP spectrum, the 0.2 and 1.0sec spectral accelerations are the only ones for which the return periods can be definitively stated. The unquantified return periods of the spectral accelerations at other periods inherently violates the underlying concepts of PBEE.
PARABOLIC DESIGN SPECTRUM
We examined several alternative spectral shapes with the goal of selecting one that had a relatively simple functional form and that reasonably represented a uniform probability of exceedance across all periods (i.e., uniform hazard spectrum, UHS). It was observed by the authors that when plotted on loglog scales, UHS for rock sites are characteristically parabolic in shape (e.g., figure 5). From analytical geometry, a parabola can be defined in terms of any three points lying on it or in terms of the location of its vertex and focus, as shown in figure 6. Given that the USGS provides hazard maps for peak ground acceleration (pga) and spectral accelerations corresponding to 0.2 and 1.0sec (i.e., S_{S} and S_{1}, respectively) as shown in figure 3, these three points were chosen to define the parabolic design spectrum. However, because oscillator period is plotted on a log scale, the pga cannot correspond to an oscillator period equal to zero (i.e., T_{pga} > 0 sec). In this context, T_{pga} can be viewed as the period at which the response spectrum converges to the pga. For example, based on the analysis of several earthquake records from the western United States, Newmark and Hall (1982) approximated T_{pga} = 0.03 sec and used this value to construct their design spectrum.
From analytical geometry, the equation for the parabolic design spectrum written in terms of pga, S_{S}, S_{1}, and T_{pga} is: (4a) where: (4b) (4c) (4d)
From examination of the above expressions, it may be observed that the coefficients a and b define the shape of the spectrum and the coefficient c scales the spectrum. An alternative form of equation (4d) for coefficient c that allows the parabolic spectrum to be scaled to an arbitrary peak acceleration (pga′) is: (4e)
Note: In using equation 4e to scale the parabolic spectrum to pga', the expression for the design spectrum becomes: (4f) (i.e., the twothirds factor is dropped from equation 4a).
COMPARISON AND DISCUSSION OF THE NEHRP AND PARABOLIC SPECTRAL SHAPES
To illustrate trends in the spectral shapes, NEHRP (2001) and parabolic spectra are computed for four cities, two in the western United States (WUS: Los Angeles and San Francisco) and two in the centraleastern United States (CEUS: Memphis, Tennessee, and Charleston, South Carolina). The hazard curves (i.e., spectral acceleration versus return period) for several oscillator periods for these cities are shown in figures 7(A)–(D). The hazard curves were drawn using data obtained from Frankel and Leyendecker (2001). From these curves, the 5% damped spectral accelerations corresponding to 0.0, 0.1, 0.2, 0.3, 0.5, 1.0, and 2.0 sec and having a return period of 2,475 years were obtained for each city and plotted in figures 8(A)–(D). Using the 0.2 and 1.0sec spectral accelerations from above in conjunction with equations (1), (2), and (3), the NEHRP (2001) spectra were computed for each city. The resulting spectra are shown in figures 8(A)–(D). Additionally, using the 0.0, 0.2, and 1.0sec spectral accelerations obtained from figure 7 in conjunction with equation (4) parabolic spectra were computed for the four cities. In using equation (4), T_{pga} values were determined such that the parabolic spectra gave the best fit of the 2,475year spectral accelerations obtained from figure 7 and plotted in figure 8. The resulting parabolic spectra are shown in figures 9(A)–(D) on loglog scales and in figures 8(A)–(D) on arithmetic scales. (Note: The NEHRP and parabolic spectra plotted in figures 8 and 9 are multiplied by 1.5. This is to remove the twothirds factor in equations (2) and (4a) because this factor relates to the inherent conservatism in the building code design procedures, not to the return period of the ground motions.)
Using the hazard curves plotted in figure 7, the return periods of the spectral accelerations defined by the NEHRP (2001) and parabolic spectra at 0.0, 0.1, 0.2, 0.3, 0.5, 1.0, and 2.0 sec were computed and are listed in table 1 for the four cities. As may be observed from the tabulated values, the return period for the NEHRP spectral accelerations ranges from 2,057 to 5,889 years, from 1,435 to 4,837 years, from 1,771 to 3,524 years, and from 1,687 to 3,239 years for Los Angeles, San Francisco, Memphis, and Charleston, respectively. In contrast, the return period for the parabolic spectral accelerations, excluding that for 2.0 sec, ranges from 2,464 to 2,561 years, from 2,227 to 2,475 years, from 2,475 to 2,637 years, and from 2,475 to 2,579 years for Los Angeles, San Francisco, Memphis, and Charleston, respectively. (Note: The 2.0sec parabolic spectral accelerations are discussed in detail below.) A comparison of these ranges shows that the variation in the return periods of the NEHRP spectral accelerations is significantly larger than that for the parabolic spectra.
Although the values in table 1 give insights into the variation of the return period of the spectral accelerations defined by the NEHRP and parabolic spectra, they do not give any insight into how overdesigned or underdesigned buildings constructed to these spectra are. To show this, the ratio of the spectral accelerations defined by the NEHRP and parabolic spectra to those having a return period of 2,475 years are listed in table 2. As may be observed from the values in table 2, the NEHRP (2001) overprescribes the 2,475year uniform hazard spectral accelerations at some periods by ∼24%, ∼15%, ∼25%, and ∼20% for Los Angeles, San Francisco, Memphis, and Charleston, respectively. At other periods the NEHRP spectra underprescribes the 2,475year spectral accelerations by ∼15%, ∼11%, ∼23%, and ∼27% for the same cities, respectively. In contrast, the maximum overprescription of 2,475year spectral accelerations by the parabolic spectra is 6% for all the cities, and the minimum underprescription, excluding the 0.2sec spectral accelerations, is 3% for all the cities. Consequently, the probabilities of “failure” of buildings designed using the NEHRP spectra vary significantly depending on the natural periods of the buildings, while buildings designed using the parabolic spectra will have similar probabilities of “failure,” independent of natural period.
Additional insights about the spectral shapes can be gained from the visual comparison of the NEHRP and parabolic spectra with the 2,475year spectral accelerations shown in figures 8(A)–(D). One striking observation is that NEHRP spectral shape does a fair job representing the 2,475year spectral accelerations for the two WUS cities, while it does a poorer job for the two CEUS cities. This is not altogether surprising, because the 2,475year spectral accelerations for both Los Angeles and San Francisco largely result from single earthquake scenarios (e.g., 75+% of the contribution to the 2,475year seismic hazard in Los Angeles is from an M 7.0 event at a distance of less than 25 km, and 80+% of the contribution to the 2,475year seismic hazard in San Francisco is from an M 8.0 event at a distance of less than 25 km). Consequently, the 2,475year spectral accelerations largely represent these single scenarios. In contrast, the 2,475year spectral accelerations for Memphis and Charleston result from a broader range of earthquake scenarios, and as a result, the 2,475year spectral values for these cities look less like values from single events. Additionally, the CEUS is tectonically stable, and earthquakes in this region are typically associated with low attenuation rates. Thus, the ground motions in the CEUS are richer in higher frequencies, as compared to WUS earthquake motions (e.g., Thenhaus and Campbell 2003). This is reflected in the fact that the 2,475year, 0.1sec spectral acceleration is larger than that for the 0.2sec for the two CEUS cities (figures 8C and D), while the reverse case is true for the two WUS cities (figures 8A and B).
To identify trends in the spectral shapes beyond those made for the four cities above, NEHRP and parabolic design spectra were computed for 493 cities that are geographically dispersed across the conterminous 48 states (383 CEUS cities and 110 WUS cities). The demarcation between the two regions was assumed to closely follow the eastern edge of the Basin and Range Province (Frankel et al. 2000). A list of the cities by state and longitude and latitude is presented in appendix A. For the parabolic spectra, the T_{pga} values were determined using the same approach as described above for the four example cities. Namely, the spectral accelerations corresponding to oscillator periods of 0.0, 0.1, 0.2, 0.3, 0.5, 1.0, and 2.0 sec and having a 2,475year return period were determined for the cities. Then, using equation (4), T_{pga} values were determined such that the parabolic spectra gave a best fit of the uniform hazard data; the resulting T_{pga} values are listed in appendix A. The correlation coefficients (i.e., r ^{2}) for both the NEHRP (2001) and parabolic spectra to the 2,475year spectral accelerations were computed for each city; the resulting r ^{2} values are listed in appendix A. Plots of the correlation coefficients for the parabolic spectra to the uniform hazard data versus the corresponding correlation coefficients of the NEHRP (2001) spectra to the uniform hazard data are presented in figures 10(A) and (B) for the WUS and CEUS, respectively. As may be observed from these figures, the parabolic spectra give a much better fit of the uniform hazard data than do the NEHRP (2001) spectra for both the WUS and CEUS. For the NEHRP spectra, r ^{2} ranges from 0.721 to 0.988, while for the parabolic spectra r ^{2} ranges from 0.932 to 1.00.
In the examples for the four cities presented above, a detailed discussion of the 2.0sec spectral acceleration for the parabolic spectra was deferred until now. A review of the return periods listed in table 1 shows that the parabolic spectra underprescribe the 2.0sec spectral accelerations for all four cities (i.e., return period < 2,475 years). To examine this trend further, we computed the 2.0sec spectral accelerations for both the parabolic and NEHRP spectra for the 493 cities listed in appendix A, with the results plotted in figures 11(A) and (B) for the WUS and CEUS, respectively. The trends shown in these figures confirm the trends identified previously (i.e., the parabolic spectra tends to slightly underprescribe the 2.0sec spectral acceleration, with the trending being more pronounced for the WUS than for the CEUS). One possible resolution to this shortcoming of the parabolic spectral shape is to use the parabolic shape for periods less than or equal to 1.0 sec and use a different shape for periods greater than 1.0 sec [e.g., S_{a}(T) = S_{1}/T for T > 1.0 sec]. However, further work is required before the authors can recommend this resolution.
Finally, the parabolic spectra for Los Angeles and San Francisco (figures 8A and B, respectively) show that the peak spectral accelerations occur at periods greater than or equal to 0.2 sec. In contrast, the peak parabolic spectral accelerations for Memphis and Charleston (figures 8C and D, respectively) occur at periods less than 0.1 sec. The main concern with this trend is that the parabolic spectra may unknowingly overprescribe spectral accelerations between T_{pga} and 0.1 sec for CEUS cities (i.e., when the peak parabolic spectral acceleration occurs at a period between T_{pga} and 0.1 sec, it may be larger than all of the known spectral accelerations). To examine this trend further, we computed the period (T_{max}) of the peak parabolic spectral accelerations for the 493 cities listed in appendix A. Histograms of the T_{max} values for the WUS and CEUS are plotted in figures 12(A) and (B), respectively. As may be observed from these histograms, T_{max} for all 110 WUS cities are greater than 0.1 sec and a little more than half of the 383 CEUS cities (202 cities) are less than 0.1 sec. The significance of this issue is open to discussion, and further research is required to examine the period corresponding to the maximum spectral acceleration for uniform hazard spectra in the CEUS. However, this research is inherently hindered by the dearth of predictive equations for spectral accelerations for the CEUS for periods between 0.0 sec and 0.1 sec.
CONTOUR MAP FOR T_{pga}
Before the parabolic spectrum can be adopted by building codes, guidance needs to be given regarding the calibration parameter T_{pga}. Toward this end, T_{pga} values were computed for the 493 cities listed in appendix A. Similar to the contour maps for pga, S_{S}, and S_{1} presented in figure 3, a contour map was developed for T_{pga} and is presented in figure 13. As may be observed from this map, there is a clear distinction in the T_{pga} values for the WUS and the CEUS; as stated previously, the demarcation between the two regions closely follows the eastern edge of the Basin and Range Province (Frankel et al. 2000). The T_{pga} values for the CEUS are characteristically lower than those in the WUS. This trend is related to the tectonic environment of the two regions. As stated previously, the CEUS is tectonically stable, and earthquakes in this region are typically associated with low attenuation rates. Thus, the ground motions in the CEUS are richer in higher frequencies, as compared to WUS earthquake motions (e.g., Thenhaus and Campbell 2003). The distinction between the T_{pga} values for the CEUS and WUS is highlighted in the histogram shown in figure 14. As may be observed from this figure, the data show a clear bimodal distribution, with the T_{pga} values for the CEUS forming the distribution on the lefthand side of the plot (shorter period range) and the T_{pga} values for the WUS forming the distribution on the righthand side of the plot (longer period range). Interestingly enough, the separation of the two distributions occurs at ∼0.03 sec, which is approximately the T_{pga} value used by Newmark and Hall (1982) to construct their design spectrum.
CONCLUSIONS
In line with performancebased earthquake engineering concepts, the authors propose a relatively simple spectral shape that reasonably represents a uniform probability of exceedance across all periods, particularly for periods less than 1.0 sec. The proposed spectral shape is parabolic when plotted on logarithmic scales. The parabolic design spectrum for firm rock sites is constructed from pga, S_{S}, S_{l}, and T_{pga}. The USGS developed contour maps for the United States for the first three parameters, and the authors developed a contour map for T_{pga}. The correlation coefficients for the parabolic spectra fit to USGS 1997 uniform hazard values for approximately 500 cities range from 0.932 to 1.00, while the correlation coefficients of the NEHRP (2001) spectra to the same uniform hazard data range from 0.721 to 0.988. These ranges confirm that the parabolic spectral shape better represents a uniform probability of exceedance across all periods than the NEHRP (2001) spectral shape. However, additional work is required to determine how to best modify the parabolic spectral shape at periods greater than 1.0 sec. Finally, in subsequent papers, factors for adjusting the firm rock parabolic spectrum to other NEHRP site classes and damping levels other than 5% of critical will be presented and discussed.
Acknowledgments
The senior author gratefully acknowledges Dr. E.V. Leyendecker for providing a copy (Frankel and Leyendecker 2001). Also, a significant portion of the work for this paper was performed as a series of undergraduate research projects and semester projects for various courses at the University of Michigan. In the case of the former, the senior author gratefully acknowledges student support received from the Marian Sarah Parker Undergraduate Scholarship Program and the College of Engineering Summer Undergraduate Research Award Program. Finally, the authors thank the two anonymous reviewers and Dr. Martin Chapman for their insightful and constructive comments.
Footnotes

University of Michigan