## 4.1 Critical Parameters and Their Ranges

_{gx0}and a

_{gy0}are the peak values of the x and y components, respectively, of the ground acceleration; and \(\ddot{\overline{u}}_{gx} \left( t \right) = {{\ddot{u}_{gx} \left( t \right)} / {a_{gx0} }}\) and \(\ddot{\overline{u}}_{gy} \left( t \right) = {{\ddot{u}_{gy} \left( t \right)} / {a_{gy0} }}\) represent the waveforms of the ground acceleration history. Equations (4.4) and (4.5) imply that the normalized displacements, namely, \(\overline{u}_{rx} \left( t \right)\), \(\overline{u}_{ry} \left( t \right)\), \(\overline{u}_{sx} \left( t \right)\) and \(\overline{u}_{sy} \left( t \right)\), are dependent only on \(\omega_{x} ,\omega_{y} /\omega_{x} ,\xi_{x} ,\xi_{y} ,\alpha ,\mu_{s} /\mu ,a_{gx0} /\mu g,a_{gy0} /a_{gx0}\), \(\ddot{\overline{u}}_{gx} \left( t \right)\), \(\ddot{\overline{u}}_{gy} \left( t \right)\) and \({{\ddot{u}_{gz} \left( t \right)} / g}\), among which a

_{gy0}/a

_{gx0}, \(\ddot{\overline{u}}_{gx} \left( t \right)\), \(\ddot{\overline{u}}_{gy} \left( t \right)\) and \({{\ddot{u}_{gz} \left( t \right)} / g}\) are associated with the ground motion characteristics.

_{x}, where A

_{x}is the peak pseudoacceleration (Chopra, 2001) in the x-direction, which is defined as

_{x}/μg, can be written as

_{x}/μg is equivalent to the maximum absolute value of the normalized relative displacement, \(\overline{u}_{rx} \left( t \right)\). Since \(\overline{u}_{rx} \left( t \right)\) is dependent only on \(\omega_{x} ,\omega_{y} /\omega_{x} ,\xi_{x} ,\xi_{y} ,\alpha ,\mu_{s} /\mu ,a_{gx0} /\mu g,a_{gy0} /a_{gx0}\), \(\ddot{\overline{u}}_{gx} \left( t \right)\), \(\ddot{\overline{u}}_{gy} \left( t \right)\) and \({{\ddot{u}_{gz} \left( t \right)} / g}\), A

_{x}/μg is also only dependent on these parameters. According to the principle of symmetry, the parameters that determine the response in the y-direction are the same as those that determine the response in the x-direction.

_{x}= 2π/ω

_{x}and T

_{y}= 2π/ω

_{y}in the x-direction and y-direction, respectively, are limited under 1.0 s. Building structures are typically designed with similar stiffnesses in two orthogonal directions; therefore, the range of T

_{x}/T

_{y}= ω

_{y}/ω

_{x}, the ratio of superstructure periods in the two horizontal directions, is assumed to be from 1/2 to 2. For the general application of SB isolation in masonry structures (e.g., Nanda et al., 2015; Qamaruddin et al., 1986), bond beams are constructed under the masonry walls as the SB element; since the bond beams weigh less than the roof (or floor) diaphragm, the resulting mass ratio, α, will be larger than 0.5 (Qamaruddin et al., 1986). When using sliding isolation bearings (e.g., Jampole et al., 2016), the mass of every floor is considered nearly equal; thus, the resulting mass ratio is approximately 0.5 for single-story buildings and over 0.5 for multistory buildings. Based on the statements provided, the mass ratio, α, is taken to be not less than 0.5 for subsequent analyses. Several studies (Barbagallo et al., 2017; Nanda et al., 2012, 2015; Yegian et al., 2004) investigating friction characteristics of sliding interfaces have found that the static friction coefficient, μ

_{s}, is slightly greater than the dynamic friction coefficient μ; previous studies (Yegian et al., 2004) found that the maximum observed value of μ

_{s}/μ was 1.38. Therefore, μ

_{s}/μ is taken to range from 1.0 to 1.4. The damping ratios in the x and y directions (i.e., ξ

_{x}and ξ

_{y}, respectively) are both taken as 5%, which is a commonly adopted value. The range of the values of the dynamic friction coefficient, μ, of the sliding interfaces investigated for SB structures (Barbagallo et al., 2017; Hasani, 1996; Jampole et al., 2016; Nanda et al., 2012; Qamaruddin et al., 1986; Yegian et al., 2004) is from 0.07 to 0.41. Furthermore, since the peak ground acceleration (PGA) seldom exceeds 1.2g, the maximum value considered for a

_{gx0}/μg is taken as 20, i.e., a

_{gx0}= 1.4g if μ = 0.07.

## 4.2 Earthquake Ground Motions Considered

_{gx0}, is not less than 0.15g were considered. However, less than 50 records with a

_{gx0}≥ 0.05g could be found in the PEER-NGA database for site classes B and E; as a result of this limitation, 40 acceleration records were selected for site classes B and E, each with a

_{gx0}≥ 0.05g, and all of them were non-pulse-like records. Site classes C and D each had 120 acceleration records selected; within those 120 records, there were 90 non-pulse-like records and 30 near-fault pulse-like records in each group. The 90 non-pulse-like records for both site class C and site class D were selected according to different combinations of the magnitude interval and source-to-site distance (defined as the closest distance to the fault rupture zone) interval. Three intervals of the magnitude, M, namely, 5.2 ≤ M < 6.0, 6.0 ≤ M < 6.7 and 6.7 ≤ M < 7.7, and three intervals of the source-to-site distance, D, namely, 0 < D < 14 km, 14 ≤ D < 24 km and 24 ≤ D < 120 km, were employed herein, resulting in 9 different combinations of magnitude and distance intervals. Given a specific site class (C or D), 10 records were selected for each of these combinations. A total of 320 records of earthquake ground motion, originating from 69 earthquakes with a magnitude M ranging from 5.2 to 7.7, were used in this study. Figure 4.2 shows the distribution of the magnitudes and source-to-site distances of the ground motion records selected for each site class.

_{gy0}/a

_{gx0}and a

_{gz0}/a

_{gx0}for the 320 selected ground motion records. As shown in Fig. 4.3a, the values of a

_{gy0}/a

_{gx0}are mostly (95.6%) between 0.5 and 2 with an average value of 1.02. This shows that the assessed earthquake ground motions typically demonstrate comparable intensities in both horizontal directions. Figure 4.3b reveals that the a

_{gz0}/a

_{gx0}values are primarily concentrated within the 0.2–0.8 range, with an average value of 0.62, which indicates that the PGA of the vertical component is generally smaller than those of the horizontal component.

## 4.3 Normalized Ground Motion Intensity for the Initiation of Sliding

_{y}is taken to be the same as T

_{x}. Figure 4.4a shows that as the mass ratio, \(\alpha\), decreases, the median value of \(\max \left( {a_{gx0} ,a_{gy0} } \right)/\mu_{cr} g\) increases for a given superstructure period, T

_{x}. The reason for this is that, in short-period structures, the peak relative acceleration is typically larger than the corresponding PGA, leading to larger values of \(\mu_{cr}\) in Eq. (4.8) for larger values of \(\alpha\). Figure 4.4a also shows that, except for T

_{x}≤ 0.3 s, the median value of \(\max \left( {a_{gx0} ,a_{gy0} } \right)/\mu_{cr} g\) increases as T

_{x}increases for a given mass ratio. This result is consistent with the shape of the corresponding response spectrum of FB structures. When \(\max \left( {a_{gx0} ,a_{gy0} } \right)/\mu_{s} g\) is equal to the corresponding counted 5th percentile value of \(\max \left( {a_{gx0} ,a_{gy0} } \right)/\mu_{cr} g\), sliding occurs for a small number of ground motions. However, the effect of this short-term sliding on the superstructure response is insignificant because of the very short sliding duration in these cases. Therefore, the response of an SB structure can be considered the same as that of the corresponding FB structure when \(\max \left( {a_{gx0} ,a_{gy0} } \right)/\mu_{s} g\) is smaller than the corresponding value in Fig. 4.4b. During 0.2 s ≤ T

_{x}≤ 0.4 s, the 5th percentile value of \(\max \left( {a_{gx0} ,a_{gy0} } \right)/\mu_{cr} g\) is estimated to be 0.27 for \(\alpha = 0.9\), and it increases to 0.42 when \(\alpha\) = 0.5. The values in Fig. 4.4c can be regarded as the lower bounds of \(\max \left( {a_{gx0} ,a_{gy0} } \right)/\mu_{s} g\) to ensure the occurrence of sliding. The trends observed in Fig. 4.4 are also evident in the results for the other site classes, except with slight difference in their respective specific values.

## 4.4 Parametric Study for the Maximum Superstructure Response

_{gx0}/μg, ten levels of a

_{gx0}/μg, namely, 0.25, 0.5, 1, 2, 4, 6, 8, 12, 16 and 20, in which a

_{gx0}/μg = 0.25 is basically equivalent to the FB case, are used in the following analyses. The dynamic friction coefficient, μ, for the sliding interfaces used in SB structures (Barbagallo et al., 2017; Hasani, 1996; Jampole et al., 2016; Nanda et al., 2012; Qamaruddin et al., 1986; Yegian et al., 2004) falls between 0.07 and 0.41. If μ is limited to the range between 0.07 and 0.41, none of the selected ground motion records can yield all the levels of a

_{gx0}/μg considered. However, in order to analyze the dispersion of the superstructure response at various levels of a

_{gx0}/μg, it is necessary to apply the same number of ground motion records for all a

_{gx0}/μg levels. For this purpose, the value of μ is adjusted with unscaled ground motion records for each target value of a

_{gx0}/μg. By doing so, to reach a large value of a

_{gx0}/μg (e.g., a

_{gx0}/μg = 16 or 20) for ground motion records with small PGAs, it will be inevitable to use very small values of μ (e.g., μ ≤ 0.02).

_{s}= \(\mu\) and T

_{x}= T

_{y}= 0.3 s are adopted. The mean and the counted median agree well with each other, as do the mean plus SD and the counted 84th percentile. As stated above, the value of \(\mu\) is adjusted to reach the target value of a

_{gx0}/\(\mu\)g with unscaled ground motion records. The considered cases are classified into two groups based on the values of \(\mu\) obtained; one with \(\mu\) within the common range of 0.07–0.41 and the other with μ outside this range. Different symbols are used in Fig. 4.5 to denote the data in these two groups. As expected, the resulting values of \(\mu\) are basically beyond the common range when a

_{gx0}/\(\mu\)g ≥ 12, because there are very few records of PGA ≥ 0.8g in the ground motion database. In order to determine the reliability of the results obtained using μ which is out of the common range, the probability densities of the computed values of A

_{x}/μg at a

_{gx0}/μg = 1 and 4 corresponding to the two different groups are compared in Fig. 4.6; in this figure, the normal probability density functions with the corresponding mean and SD are also presented. The distributions of \(A_{x} /\mu g\) in each group are fundamentally similar to each other. Based on this observation, similar results would likely be obtained for large values of a

_{gx0}/μg if a sufficient number of ground motion records with large PGAs were used. Additionally, the probability density of the calculated values corresponds quite well with the corresponding fitted normal probability density function, which suggests that the probability distribution of \(A_{x} /\mu g\) for a given \(a_{gx0} /\mu g\) value is approximately in accordance with a normal distribution.

### 4.4.1 Comparison of the Response in Two Orthogonal Directions

_{s}= μ and T

_{x}= T

_{y}= 0.3 s are used. Figure 4.7 shows a basically uniform distribution of the discrete points with \(\left( {a_{gy0} /\mu g,A_{y} /\mu g} \right)\) coordinates along both sides of the mean \(A_{x} /\mu g\) versus \(a_{gx0} /\mu g\) curve. This indicates that the normalized peak pseudoacceleration and the normalized PGA have an essentially identical relationship in both orthogonal horizontal directions; in other words, the results obtained for the x-direction can also be applied to the y-direction. Therefore, only the response in the x-direction is analyzed hereafter.

### 4.4.2 Effect of the Vertical Earthquake Component

_{x}under three-component excitation to that under the corresponding excitation with the two horizontal components. The 90 non-pulse-like records selected for site class D with μ

_{s}= μ and T

_{x}= T

_{y}were used to compute the results shown in Fig. 4.8. According to these figures, the vertical component of ground motion can either increase or decrease the horizontal response of the superstructure. In general, the vertical component has a greater effect on stiffer structures. For certain ground motions, with T

_{x}≤ 0.3 s, the ratios of A

_{x}under three-component excitation to that under two-component excitation exceed 1.15. However, for most of the ground motions considered, the ratios of A

_{x}under three-component excitation to that under two-component excitation are between 0.95 and 1.05, and the mean values are basically equal to 1.0. Therefore, the overall effect of the vertical component on the superstructure response is negligible.

### 4.4.3 Effect of the Natural Period of the Superstructure

_{x}/μg and T

_{x}for different values of \(\alpha\) and \(a_{gx0} /\mu g\). The mean values of \(A_{x} /\mu g\) were computed using the 90 non-pulse-like records for site class D while assuming that μ

_{s}= μ and T

_{x}= T

_{y}. According to Fig. 4.9, as T

_{x}increases, the mean values of \(A_{x} /\mu g\) increase for T

_{x}≤ 0.3 s, but decrease for T

_{x}≥ 0.4 s, resulting in the maximum mean values of \(A_{x} /\mu g\) are obtained at T

_{x}= 0.3 s or 0.4 s. In general, the mean values of \(A_{x} /\mu g\) at T

_{x}= 0.3 s or 0.4 s are close to each other. As \(a_{gx0} /\mu g\) increases, the influence of T

_{x}on the superstructure response decreases; for example, the ratio of the minimum to the maximum mean values of \(A_{x} /\mu g\) in Fig. 4.9a is 0.69 for \(a_{gx0} /\mu g\) = 2 and increases to 0.87 for \(a_{gx0} /\mu g\) = 12. As shown in Fig. 4.9, it can be inferred that the mean value of \(A_{x} /\mu g\) does not exhibit a considerable variation within the range of T

_{x}that has been considered; therefore, it is appropriate to use the response from the period with the maximum mean \(A_{x} /\mu g\) as a representation of the responses for possible SB structures. As mentioned above, for site class D, this period can be taken as 0.3 s; for site classes B, C, and E, the critical periods obtained are 0.2, 0.2, and 0.4 s, respectively.

_{x}= T

_{y}is adopted. Figure 4.10 compares the mean values of \(A_{x} /\mu g\) for different values of T

_{x}/T

_{y}in order to investigate the possible impact of T

_{x}/T

_{y}on the superstructure response. In general, the mean values of \(A_{x} /\mu g\) at a given level of \(a_{gx0} /\mu g\) decreases as T

_{x}/T

_{y}increases. Nevertheless, this variation is quite limited; the presented results in Fig. 4.10 show that the ratios of the mean value of \(A_{x} /\mu g\) for T

_{x}/T

_{y}= 0.5 to T

_{x}/T

_{y}= 2 do not exceed 1.06. Therefore, the results obtained for T

_{x}= T

_{y}are representative of those obtained for the possible value of T

_{x}/T

_{y}in the range considered.

### 4.4.4 Effect of the Difference Between the Static and Dynamic Friction Coefficients

_{s}/\(\mu\) generally within the range of 1.0–1.4. Figure 4.11 compares the mean values of A

_{x}/μg corresponding to different values of μ

_{s}/μ in order to investigate the influence of μ

_{s}/μ. As expected, for most of the ground motions considered, the superstructure response is not significantly influenced by the value of μ

_{s}/μ when a

_{gx0}/μg = 0.25, because sliding does not occur at this a

_{gx0}/μg level. The influence of μ

_{s}/μ is most clearly observed for a

_{gx0}/μg = 1. If a

_{gx0}/μg exceeds 1, the influence of μ

_{s}/μ decreases as a

_{gx0}/μg increases because for larger a

_{gx0}/μg values, the responses of SB structures are primarily dominated by the sliding phase, during which the responses are independent of the static friction coefficient. The ratios of the mean value of A

_{x}/μg for μ

_{s}/μ = 1.4 to that for μ

_{s}/μ = 1 are all below 1.08 and mostly below 1.02. Therefore, the effect of the difference between the static and dynamic friction coefficients can be neglected. In the following analyses, μ

_{s}= μ is assumed.

### 4.4.5 Effects of the Earthquake Magnitude and Source-to-Site Distance

_{x}/μg for each distance interval and magnitude were computed using 90 non-pulse-like records selected for site class D in order to study the effects of the earthquake magnitude and source-to-site distance. Figure 4.12 shows the mean values of \(A_{x} /\mu g\) for the three magnitude intervals. For the entire range of \(a_{gx0} /\mu g\), no general trend can be observed for \(A_{x} /\mu g\) as the earthquake magnitude increases. However, when \(a_{gx0} /\mu g\) = 0.25, the mean values of \(A_{x} /\mu g\) for 6.7 ≤ M < 7.7 are approximately 10% larger than those for 6.0 ≤ M < 6.7; the relative differences in the mean values of \(A_{x} /\mu g\) between any two of these groups at a given \(a_{gx0} /\mu g\) are all below 7% and mostly below 5%. This indicates that the earthquake magnitude has little effect on the superstructure responses of SB structures.

### 4.4.6 Effect of Near-Fault Pulses

_{x}/a

_{gx0}, for non-pulse-like ground motions compared to pulse-like ones, as indicated. Chopra and Chintanapakdee (2001) reported similar results, where they examined the normalized response spectra of harmonic excitations containing different numbers of cycles to interpret this phenomenon. The response amplification factor increased as the number of cycles increased, implying that the response amplification factors of pulse-like ground motions with one or several dominant pulses are generally smaller than those of non-pulse-like ground motions with more excitation cycles. As illustrated in Fig. 4.14a, when \(\alpha\) = 0.5, the ratio of the mean value of \(A_{x} /\mu g\) for the non-pulse-like records to that of the pulse-like records remains almost the same as \(a_{gx0} /\mu g\) increases. However, when \(\alpha\) increases to 0.8, the ratio decreases significantly as \(a_{gx0} /\mu g\) increases, as shown in Fig. 4.14b. This phenomenon is consistent with the fact that the difference in the superstructure response between different ground motions decreases as \(\alpha\) increases. More detailed results related to this fact are presented in the next section.

### 4.4.7 Statistical Results for Each Site Class

_{rx}as \(\alpha\) increases.

_{gx0}/\(\mu\)g curves experience a rapid decline in tangent slopes as a result of sliding. When \(a_{gx0} /\mu g\) exceeds a sufficiently large value, i.e., there is an upper limit for the superstructure response of an SB structure, the tangent slopes are expected to finally become 0. This situation is favorable for isolating extremely large earthquakes. The efficiency of the SB system can also be demonstrated by the value of A

_{x}/a

_{gx0}, which is equal to the origin-oriented secant slope of the \(A_{x} /\mu g\) versus \(a_{gx0} /\mu g\) curve. Taking \(\alpha = 0.8\) in Fig. 4.15b (site class C) as an example: when \(a_{gx0} /\mu g\) = 2, the mean value of A

_{x}/a

_{gx0}is 0.82, whereas this value is 2.23 for the FB case; consequently, the superstructure response of the SB structure is just 36.8% of that of the corresponding FB structure in this instance.

_{x}/μg increase. For a

_{gx0}/μg = 0.25, when sliding basically does not occur, the ratios of the mean values of A

_{x}/μg for site classes C, D and E to that for site class B are equal to 1.19, 1.20 and 1.24, respectively. These ratios generally decrease as a

_{gx0}/μg increases. As shown in Fig. 4.16a, when \(a_{gx0} /\mu g\) = 2, these ratios decrease to 1.07, 1.07 and 1.12 for site classes C, D and E, respectively; and they further decrease to 1.02, 1.03 and 1.04 when \(a_{gx0} /\mu g\) = 12.

_{x}/μg versus a

_{gx0}/μg curves are similar for all four site classes. In the range of a

_{gx0}/μg ≤ 2, the COVs of A

_{x}/μg decrease rapidly as a

_{gx0}/μg increases; after a

_{gx0}/μg exceeds 2, the COVs of A

_{x}/μg are quite constant. This means that sliding tends to reduce the dispersion of the superstructure response due to the record-to-record variability. As \(\alpha\) increases, the COVs of \(A_{x} /\mu g\) decrease in general. This can also be interpreted by using Eq. (4.9); larger values of \(\alpha\) leads to larger equivalent damping ratios for the sliding phases, further resulting in smaller dispersion of the structural response.

## 4.5 Simplified Equations for Estimating the Maximum Superstructure Response

_{x}/T

_{y}, \(\mu\)

_{s}/\(\mu\), the earthquake magnitude and the source-to-site distance can be neglected; and (3) the response of possible SB structures can be represented by the response at a critical period for each site class with appropriate conservativeness. Thus, the following equation was proposed to estimate the mean values of \(A_{x} /\mu g\) and \(A_{y} /\mu g\):

Site class | \(\alpha\) | \(\beta\) _{1} | \(\beta\) _{2} | \(\beta\) _{3} |
---|---|---|---|---|

B | 0.5 | 3.26 | 1.07 | 1.12 |

0.6 | 3.20 | 0.92 | 1.22 | |

0.7 | 3.10 | 0.81 | 1.30 | |

0.8 | 2.80 | 0.73 | 1.24 | |

0.9 | 2.41 | 0.58 | 1.15 | |

C | 0.5 | 3.32 | 1.03 | 0.90 |

0.6 | 3.17 | 0.88 | 0.97 | |

0.7 | 2.92 | 0.76 | 0.97 | |

0.8 | 2.71 | 0.63 | 1.02 | |

0.9 | 2.48 | 0.45 | 1.10 | |

D | 0.5 | 3.37 | 1.01 | 0.92 |

0.6 | 3.09 | 0.90 | 0.90 | |

0.7 | 2.93 | 0.76 | 0.97 | |

0.8 | 2.78 | 0.61 | 1.06 | |

0.9 | 2.39 | 0.47 | 0.99 | |

E | 0.5 | 3.27 | 1.16 | 0.81 |

0.6 | 3.00 | 1.06 | 0.80 | |

0.7 | 2.73 | 0.94 | 0.81 | |

0.8 | 2.68 | 0.68 | 0.99 | |

0.9 | 2.87 | 0.40 | 1.41 |

_{x}/μg with respect to a specific site class, a

_{gx0}/μg and α. Therefore, if we can further derive a simplified equation for the COV (or SD = mean × COV), then the value of A

_{x}/μg (or A

_{y}/μg) corresponding to any probability of nonexceedance can be readily determined. Figure 4.17 shows that, despite possible variations of the exact values among different groups, the trends of the COVs of \(A_{x} /\mu g\) with respect to \(a_{gx0} /\mu g\) remain consistent for the four site classes. Because the computed COV values are related to the selected ground motion records used in the computation (i.e., a different set of records for the same site class may lead to different COVs), it is reasonable to expect equivalent dispersion levels for the four site classes, provided that a sufficient number of records are selected for each group. In general, the computed COV values are the largest for site class C under the circumstance of the ground motion records considered. Therefore, the data of site class C are used to derive the simplified equation for the COVs of A

_{x}/μg and A

_{y}/μg since a larger COV value leads to a conservative result for a probability of nonexceedance larger than 50%. The proposed equation is given by

\(\alpha\) | \(\gamma_{1}\) | \(\gamma_{2}\) | \(\gamma_{3}\) |
---|---|---|---|

0.5 | 0.23 | 1.82 | 0.20 |

0.6 | 0.28 | 2.25 | 0.19 |

0.7 | 0.34 | 2.25 | 0.15 |

0.8 | 0.42 | 2.62 | 0.12 |

0.9 | 0.52 | 3.23 | 0.10 |

## 4.6 Conclusions

_{x}≤ 0.3 s, for certain ground motions, the superstructure response can increase by more than 1.15 due to the vertical component; but the effect of the vertical component is negligible for the majority of situations. The normalized peak pseudoacceleration exhibits a pattern of initially increasing and subsequently decreasing as the natural period of the superstructure increases. For the range of T

_{x}considered, the variation in the mean value of A

_{x}/μg is not very significant. For simplicity and conservativeness, the response of possible SB structures can be represented by the response at the period where the maximum mean A

_{x}/μg is generally obtained. The influence of the natural period ratio in the two orthogonal horizontal directions and the possible difference between the static and dynamic friction coefficients on the superstructure responses of SB structures is insignificant.

_{x}/μg with respect to a

_{gx0}/μg is similar among the four site classes. The COVs decline rapidly at smaller values of a

_{gx0}/μg and remain basically constant after a

_{gx0}/μg ≥ 2. For a given site class, a

_{gx0}/μg and α, a normal distribution is appropriate for modeling the probability distribution of A

_{x}/μg.

_{x}/μg decrease as α increases. An upper limit for the superstructure response exists for every mass ratio, which is beneficial for the isolation of extremely large earthquakes. Implementing Eqs. (4.10) and (4.11) with the associated values of the regression coefficients can provide good estimates for the mean values and COVs, respectively, of A

_{x}/μg and can be used to predict the value of the normalized peak pseudoacceleration corresponding to any probability of nonexceedance.