## 6.1 Selection of Ground Motion Intensity Measure

_{sx0}and u

_{sy0}, can be directly determined once the response histories of the sliding displacements are obtained through response history analyses. The maximum of PSDs over all the horizontal directions or the PSD with respect to the origin, u

_{st0}, can also be determined by

_{sx0}and u

_{sy0}. Generally, when using sliding isolation bearings (e.g., Jampole et al., 2016), the boundary shape of the sliding surface is circular. For this case, design requires comparison of u

_{st0}with the sliding displacement threshold. The analyses that follow investigate all of u

_{sx0}, u

_{sy0}, and u

_{st0}, covering the two cases mentioned before.

_{x}= T

_{y}= 0.4 s (T

_{x}= 2π/ω

_{x}and T

_{y}= 2π/ω

_{y}), ξ

_{x}= ξ

_{y}= 5%, and α = 0.7 subjected to the three components of the 180 ordinary ground motion records were conducted. Figure 6.1 displays the calculated values of u

_{sx0}and u

_{st0}in relation to their respective PGAs (a

_{gx0}and a

_{gt0}) and PGVs (v

_{gx0}and v

_{gt0}), where two levels of friction coefficient μ are considered, namely μ = 0.1 and 0.2. In order to maintain consistency with the definition of u

_{st0}, Eqs. (6.2) and (6.3) are respectively used to compute the PGA and PGV corresponding to u

_{st0}, which are the maximum values of PGA and PGV over all the horizontal directions.

_{s}, for nonlinear correlations is used here. For μ = 0.1, the computed values of ρ

_{s}are 0.38 (0.38) and 0.80 (0.85) for the correlations between u

_{sx0}and a

_{gx0}(u

_{st0}and a

_{gt0}), and u

_{sx0}and v

_{gx0}(u

_{st0}and v

_{gt0}), respectively; and for μ = 0.2, these values are 0.56 (0.60) and 0.69 (0.73), respectively. The efficiency of PGV as an IM is relatively high, and it improves with increased sliding extent, which is appreciated because design is mainly concerned with sliding displacements that are sufficiently large and may exceed the preset threshold. Nevertheless, the variability of the PSD at a given PGV is still considerable in comparison with the peak superstructure response of SB structures presented in Chap. 4. This relatively large variability is primarily attributed to the following reasons, which have been pointed out by Jampole et al. (2020):

## 6.2 Critical Parameters and Their Ranges

_{gx0}/μ and v

_{gy0}/μ (where v

_{gx0}and v

_{gy0}are the PGVs in the x and y directions, respectively), and μ is not an independent variable anymore. Thus, to simplify the estimation of PSDs related to different levels of ground motion intensity associated with various levels of μ, we can evaluate the normalized PSDs (u

_{sx0}/μ, u

_{sy0}/μ, and u

_{st0}/μ) at various normalized PGVs (v

_{gx0}/μ, v

_{gy0}/μ, and v

_{gt0}/μ). Equations (6.4)–(6.6) shows that ω

_{x}, ω

_{y}, ξ

_{x}, ξ

_{y}, α, \(\ddot{u}_{gz} \left( t \right)\), and the horizontal ground motion waveform are other parameters that may affect the normalized PSDs.

_{x}= 2π/ω

_{x}and T

_{y}= 2π/ω

_{y}), the damping ratios (ξ

_{x}and ξ

_{y}), and the mass ratio (α) have been presented in Sect. 4.2, and thus are not repeated here. The friction coefficients 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) range from 0.07 to 0.41. Apart from very few near-fault records from high-magnitude earthquakes, most of the ground motions recorded have PGVs below 1.2 m/s. On these bases, the normalized PGV is limited to 18 m/s, which is equivalent to PGV = 1.26 m/s when μ = 0.07. When the normalized PGV is 1 m/s, the PSDs associated with μ lying in the common range are well below 0.1 m, a value that can be considered as the lower bound of the sliding displacement threshold in practice. Therefore, analyzing the cases with normalized PGV below 1 m/s is not necessary from a design perspective. In the following parametric study, eleven levels of normalized PGVs, namely 1, 1.5, 2, 4, 6, 8, 10 12, 14, 16, and 18 m/s, are considered. These levels of normalized PGVs were achieved by adjusting the value of μ with the ground motion records unscaled.

## 6.3 Parametric Study for the Normalized Peak Sliding Displacements

### 6.3.1 Comparison of the Responses in the Two Orthogonal Directions

_{sx0}/μ at each level of v

_{gx0}/μ considered and the data points, (v

_{gy0}/μ, u

_{sy0}/μ), corresponding to the response in the y direction. These data were obtained from response history analyses of SB structures with T

_{x}= T

_{y}= 0.4 s and α = 0.7 using the 180 non-pulse-like ground motion records. Figure 6.3 presents only the data points with 1 m/s ≤ v

_{gy0}/μ ≤ 18 m/s, in order to maintain consistency with the range of the normalized ground motion IM considered in the x direction. It was found that a quadratic polynomial curve can well represent the relationship between mean u

_{sx0}/μ and v

_{gx0}/μ; therefore, a regression curve, obtained through the use of a quadratic polynomial equation for fitting the data points, is displayed in Fig. 6.3. As can be seen in this figure, the regression curve for the relationship between the mean u

_{sy0}/μ and v

_{gy0}/μ agrees well with the curve of the mean u

_{sx0}/μ versus v

_{gx0}/μ. Therefore, it can be inferred that the relationship between the mean normalized PSD and normalized PGV is essentially identical for the two orthogonal horizontal directions. However, it is important to note that the PSDs may vary greatly between the two orthogonal directions for an individual ground motion, despite both directions having the same PGVs. In terms of statistical results, the outcomes achieved for the x-direction through a considerable number of ground motions can be extended to the y-direction. For this reason, only the response in the x direction is investigated hereafter.

### 6.3.2 Probability Distribution of the Normalized PSD at a Given Level of Normalized Ground Motion Intensity

_{sx0}/μ and u

_{st0}/μ) at four distinct levels of normalized PGV (v

_{gx0}/μ and v

_{gt0}/μ). The data used to determine these empirical cumulative distributions were derived from response history analyses using the 180 non-pulse-like ground motion records with the structural parameters T

_{x}= T

_{y}= 0.4 s and α = 0.7. The figure clearly depicts that the empirical distributions are asymmetrical around the sample median and have lengthier tails moving towards upper values. The lognormal distribution, which has been extensively utilized in seismic performance assessment of building structures (e.g., Ruiz-Garcia & Miranda, 2007; Zareian & Krawinkler, 2007), also presents such type of feature and, thus, could be suitable for modeling the probability distributions of u

_{sx0}/μ at a given level of v

_{gx0}/μ and u

_{st0}/μ at a given level of v

_{gt0}/μ. The sample geometric mean and sample logarithmic standard deviation (Ang & Tang, 2006) are typically used to estimate the two parameters (i.e., the median and logarithmic standard deviation) of the fitted lognormal distribution function. For this study, the equations for estimating the parameters can be written as

_{sx0}/μ)

_{m}and (u

_{st0}/μ)

_{m}are the medians of u

_{sx0}/μ and u

_{st0}/μ, respectively; \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\) are the lognormal standard deviations of u

_{sx0}/μ and u

_{st0}/μ, respectively; (u

_{sx0}/μ)

_{i}and (u

_{st0}/μ)

_{i}are the observed value; and n is the sample size. However, for v

_{gx0}/μ (and v

_{gt0}/μ) ≤ 2 m/s, some observed values of u

_{sx0}/μ (and u

_{st0}/μ) are 0 or very close to 0, which makes Eqs. (6.7) and (6.8) invalid because the natural logarithm of zero does not exist and the value computed by Eq. (6.7) will be dominated by the natural logarithm of a value that is near 0. Thus, for these cases, (u

_{sx0}/μ)

_{m}and (u

_{st0}/μ)

_{m}are taken as the counted medians, and \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\) are estimated by Eq. (6.9) based on the assumption that the data are sampled from lognormal distributions.

_{sx0}/μ)

_{50%}and (u

_{sx0}/μ)

_{84%}are the counted median and counted 84th percentile of u

_{sx0}/μ, respectively; and (u

_{st0}/μ)

_{50%}and (u

_{st0}/μ)

_{84%}are the counted median and counted 84th percentile of u

_{st0}/μ, respectively.

_{gx0}/μ and v

_{gt0}/μ. In general, the fitted lognormal distribution agrees fairly well with the corresponding empirical distribution. The well-known Kolmogorov–Smirnov (K-S) goodness-of-fit tests (Ang & Tang, 2006) were conducted to further verify the adequacy of the lognormal distribution. Figure 6.4 depicts the graphical representations of the K-S test with a 5% significance level. The figure displays that all data points for v

_{gx0}/μ (and v

_{gt0}/μ) = 2, 6, and 12 m/s, are within the limits of acceptability (i.e., the two dotted lines in Fig. 6.4), indicating that the assumed lognormal distribution is acceptable. For v

_{gx0}/μ (and v

_{gt0}/μ) = 1 m/s, due to the presence of several null values, certain points at the lower tail fall outside the acceptable limits; however, the practical sliding displacement threshold is much higher than the PSDs associated with this lower tail, thus the utility of the lognormal distribution remains unaffected. Concluding from the aforementioned discussions, it is evident that the lognormal distribution is appropriate for modeling the probability distributions of u

_{sx0}/μ at a given level of v

_{gx0}/μ, and u

_{st0}/μ at a given level of v

_{gt0}/μ.

### 6.3.3 Effect of the Vertical Ground Motion Component

### 6.3.4 Effects of the Superstructure Natural Period and Mass Ratio

_{sx0}/μ)

_{m}and v

_{gx0}/μ, and (u

_{st0}/μ)

_{m}and v

_{gt0}/μ, respectively, for different values of T

_{x}and α, which were determined by using the 180 non-pulse-like ground motion records and assuming T

_{x}= T

_{y}. As shown in these figures, the trend of (u

_{sx0}/μ)

_{m}changing as v

_{gx0}/μ increases closely resembles the trend of (u

_{st0}/μ)

_{m}changing as v

_{gt0}/μ increases. In comparison with the normalized PGV, the influence of T

_{x}and α on (u

_{sx0}/μ)

_{m}and (u

_{st0}/μ)

_{m}is not so significant. To further investigate the combined effects of T

_{x}and α on (u

_{sx0}/μ)

_{m}and (u

_{st0}/μ)

_{m}, (u

_{sx0}/μ)

_{m}and (u

_{st0}/μ)

_{m}are plotted against T

_{x}and α in Figs. 6.8 and 6.9, respectively, for four representative levels of normalized PGV. This figure also presents the results of rigid bodies, which correspond to T

_{x}= 0. As can be seen in Figs. 6.8 and 6.9, when T

_{x}≤ 0.4 s, the mass ratio basically has a negligible effect on (u

_{sx0}/μ)

_{m}and (u

_{st0}/μ)

_{m}; when T

_{x}> 0.4 s, the influence of α becomes slightly more significant, and (u

_{sx0}/μ)

_{m}and (u

_{st0}/μ)

_{m}generally increase as α increases. This phenomenon cannot be simply interpreted using the governing equations presented previously; additionally, since a larger value of α does not always lead to a larger (u

_{sx0}/μ)

_{m}or (u

_{st0}/μ)

_{m}, as presented in Figs. 6.8 and 6.9, the inherent characteristics of the ground motion time history should have played a significant role in this general trend. For a given mass ratio, the values of (u

_{sx0}/μ)

_{m}and (u

_{st0}/μ)

_{m}generally first increase and then decrease as T

_{x}increases, and the differences between the maximum and minimum values of (u

_{sx0}/μ)

_{m}and (u

_{st0}/μ)

_{m}for T

_{x}within the range considered range from 0.04 to 0.33 m and 0.02 to 0.27 m, respectively, and generally increase as the corresponding normalized PGV increases. From this result, we know that the PSDs of actual SB structures may be underestimated by relying solely on the response of rigid bodies. For simplicity, it is reasonable to use the maximum values of (u

_{sx0}/μ)

_{m}and (u

_{st0}/μ)

_{m}for the range of T

_{x}considered to conservatively estimate the PSDs of possible SB structures.

_{gx0}/μ, and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\) and v

_{gt0}/μ, respectively, for different values of T

_{x}and α. For some cases when v

_{gx0}/μ (and v

_{gt0}/μ) = 1 m/s, the values of (u

_{sx0}/μ)

_{50%}[and (u

_{st0}/μ)

_{50%}] are 0 or very close to 0; thus, the values obtained from using Eq. (6.9) to compute \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) [and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\)] are infinite or unreasonably large. For this reason, the results corresponding to v

_{gx0}/μ (and v

_{gt0}/μ) = 1 m/s are not presented in Fig. 6.10 (and Fig. 6.11). As shown in Fig. 6.10 (and Fig. 6.11), \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) [and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\)] generally lie between 0.4 and 0.6 except for some cases when v

_{gx0}/μ (and v

_{gt0}/μ) = 1.5 and 2 m/s. When the normalized PGV is small, sliding is not predominant, and the ground acceleration may play a more significant role than the ground velocity as the acceleration quantities dominate the initiation of sliding as revealed by Eq. (6.5); furthermore, sliding is less likely to occur for smaller values of α and larger values of T

_{x}, as demonstrated in Chap. 4. This explains why the values of \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) [and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\)] are generally larger for v

_{gx0}/μ (and v

_{gt0}/μ) = 1.5 and 2 m/s and even larger values are obtained when T

_{x}= 1 s and α ≤ 0.7.

_{x}and α on \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\) are plotted in Figs. 6.12 and 6.13, respectively, for four representative levels of normalized PGV. In general, the influence of α on \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) [and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\)] is small except for v

_{gx0}/μ (and v

_{gt0}/μ) ≤ 2 m/s. For any given level of v

_{gx0}/μ (and v

_{gt0}/μ), the maximum value of \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) [and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\)] is obtained at T

_{x}= 1 s; this value is 0.87 (and 1.12) for v

_{gx0}/μ (and v

_{gt0}/μ) = 2 m/s and is around 0.63 (and 0.58) for all other levels of v

_{gx0}/μ (and v

_{gt0}/μ) ≥ 4 m/s. The value of T

_{x}at which the minimum \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) [and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\)] is obtained varies for different levels of v

_{gx0}/μ (and v

_{gt0}/μ). For a given level of v

_{gx0}/μ (and v

_{gt0}/μ), the minimum value of \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) [and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\)] ranges from 0.41 to 0.57 (and 0.38–0.53). The average value of \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) [and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\)] is 0.60 (and 0.58) for v

_{gx0}/μ (and v

_{gt0}/μ) = 2 m/s and ranges from 0.49 to 0.58 (and 0.45–0.55) for v

_{gx0}/μ (and v

_{gt0}/μ) ≥ 4 m/s.

_{x}= T

_{y}is employed in all of the aforementioned analyses. To investigate the possible effect of T

_{x}/T

_{y}on the PSD, the values of (u

_{sx0}/μ)

_{m}and (u

_{st0}/μ)

_{m}corresponding to different values of T

_{x}/T

_{y}are compared in Figs. 6.14 and 6.15, respectively, and those of \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\) are compared in Figs. 6.16 and 6.17, respectively. For the data presented in these figures, α is taken as 0.7. These figures make it clear that T

_{x}/T

_{y}has a negligible influence. Therefore, the results obtained for T

_{x}= T

_{y}can represent those of the other T

_{x}/T

_{y}within the range considered.

### 6.3.5 Effect of Near-Fault Pulses

_{sx0}/μ)

_{m}versus v

_{gx0}/μ and (u

_{st0}/μ)

_{m}versus v

_{gt0}/μ curves obtained using the 60 near-fault pulse-like records and the 180 non-pulse-like records are compared in Figs. 6.18 and 6.19, respectively. When v

_{gx0}/μ ≤ 4 m/s (and v

_{gt0}/μ ≤ 6 m/s), the values of (u

_{sx0}/μ)

_{m}[and (u

_{st0}/μ)

_{m}] corresponding to the pulse-like records are close to those corresponding to the non-pulse-like records. When v

_{gx0}/μ exceeds 6 m/s (and v

_{gt0}/μ exceeds 8 m/s), the value of (u

_{sx0}/μ)

_{m}[and (u

_{st0}/μ)

_{m}] for the pulse-like records starts to exceed the corresponding value for the non-pulse-like records, and the difference increases monotonically as v

_{gx0}/μ (and v

_{gt0}/μ) increases. To interpret the underlying reason for this phenomenon, Fig. 6.20 (T

_{x}= T

_{y}= 0.4 s and α = 0.7 are adopted) illustrates the ground acceleration, velocity, and sliding displacement time histories corresponding to the counted median of u

_{sx0}/μ in each (non-pulse-like or pulse-like) group. By comparing the time histories presented in Fig. 6.20b for v

_{gx0}/μ = 10 m/s, we can find that the prominent long-period velocity pulse in the pulse-like ground motion is the cause of the larger value of (u

_{sx0}/μ)

_{m}in comparison with the non-pulse-like ground motion. However, when v

_{gx0}/μ is small, as illustrated in Fig. 6.20a for v

_{gx0}/μ = 2 m/s, the contribution of the long-period velocity pulse is not so significant. The simplified equation (Eq. 6.10) proposed by Jampole et al. (2018) for predicting the PSD of a rigid block subjected to a half-sine pulse which was derived from simplification of the corresponding closed-form solution can provide an approximate interpretation of this result.

_{s,max}is the PSD of the rigid block; and a

_{p}and T

_{p}are the peak acceleration and duration of the half-sine pulse, respectively. Dividing both sides of Eq. (6.10) by μ and replacing a

_{p}T

_{p}with πv

_{gi}/2 (where v

_{pi}is the peak velocity of the half-sine pulse), lead to

_{gx0}/μ and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\) versus v

_{gt0}/μ curves, respectively, of the pulse-like records with those of the non-pulse-like records. As shown in this figure, when v

_{gx0}/μ (and v

_{gt0}/μ) ≤ 4 m/s, the values of \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) [and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\)] for the pulse-like records are generally larger than those for the non-pulse-like records; when v

_{gx0}/μ (and v

_{gt0}/μ) ≥ 6 m/s, the value of \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) [and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\)] for the pulse-like records does not change much and is slightly smaller than the corresponding value for the non-pulse-like records. Since the computed dispersion is partly influenced by the selected ground motion records, and typically there are minimal differences in the computed \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) [and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\)] of the two ground motion types, it is reasonable to expect a similar level of inherent dispersion for the two types of ground motions.

## 6.4 Fragility Curves

_{x}and α on (u

_{sx0}/μ)

_{m}[and (u

_{st0}/μ)

_{m}] is limited in comparison with that of v

_{gx0}/μ (and v

_{gt0}/μ). Therefore, in the design of SB structures, it is advisable to use the maximum values of (u

_{sx0}/μ)

_{m}and (u

_{st0}/μ)

_{m}conservatively for the common range of T

_{x}. The maximum (u

_{sx0}/μ)

_{m}versus v

_{gx0}/μ and maximum (u

_{st0}/μ)

_{m}versus v

_{gt0}/μ curves are plotted in Fig. 6.23 for different values of α and for both the non-pulse-like and pulse-like ground motions. Since the curves corresponding to different values of α are very close to each other, for simplicity, equations for design can be developed solely based on the findings of α = 0.9, which are generally larger than those of other values of α. It is found that a quadratic polynomial curve can well fit the relationship between the maximum (u

_{sx0}/μ)

_{m}and v

_{gx0}/μ, as well as the maximum (u

_{st0}/μ)

_{m}and v

_{gt0}/μ, and the obtained regression formulae are as follows:

_{sx0}/μ)

_{m}and (u

_{st0}/μ)

_{m}are in m, and v

_{gx0}/μ and v

_{gt0}/μ are in m/s. According to the findings presented previously, replacing the subscript letter “x” with “y” enables the application of Eqs. (6.12a) and (6.13a) to the response in the y direction as well. As shown in Fig. 6.23, Eqs. (6.12) and (6.13) can well predict the corresponding relationships between the median normalized PSD and normalized PGV, and the coefficients of determination, R

^{2}, of these equations are all larger than 0.99. Further comparison of the curves determined by Eqs. (6.12a) and (6.12b) [and Eqs. (6.13a) and (6.13b)], as presented in Fig. 6.23, indicates that the relationship between the median normalized PSD and normalized PGV in each principal direction is close to that with respect to the origin. Since the median normalized PSD versus normalized PGV curve corresponding to Eq. (6.12b) [and Eq. (6.13a)] is slightly above that corresponding to Eq. (6.12a) [and Eq. (6.13b)], Eqs. (6.12b) and (6.13a) can be used conservatively as unified equations for predicting the response in each principal direction as well as the maximum response over all the directions.

_{st0})

_{m}, of friction pendulum isolators:

_{b}is the isolation period, and \(\eta\) is defined as

_{d}in Eq. (6.15) is the frequency marking the transition from the velocity-sensitive to the displacement-sensitive region of the median spectrum of the stronger horizontal ground-motion components. Note that in Eq. (6.14), the PSD with respect to the origin is taken as the response quantity of interest, while the PGV of the stronger component is taken as the ground motion IM, which sets it apart from the treatment in this study. The comparison of the median normalized PSD versus normalized PGV curve, determined by Eq. (6.14) [ω

_{d}= 3.05 is adopted, as done by Ryan and Chopra (2004), and T

_{b}is taken as 10 s such that the corresponding radius of the FP isolator is sufficiently large to yield the same response as that of a flat sliding system], with those determined by Eqs. (6.12) and (6.13) in Fig. 6.23, is presented. As can be seen, the curve determined by Eq. (6.14) is close to those determined by Eq. (6.13), which is proposed for near-fault pulse-like ground motions. This is because the 20 ground motions used in the response history analyses conducted by Ryan and Chopra (2004) were from large-magnitude earthquakes and recorded at sites near fault ruptures, the characteristics of which are close to the near-fault pulse-like ground motions used in the present study.

_{x}, T

_{x}/T

_{y}, and α on the logarithmic standard deviations, \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\), and the values of \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) [and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\)] are generally between 0.4 and 0.6 except for some cases when v

_{gx0}/μ (and v

_{gt0}/μ) is small. Based on these results, adopting a constant value for \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) [and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\)] is reasonable in design. This value is taken as 0.55 for both \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\) here, which is approximately the average of all the results computed using the non-pulse-like ground motion records when v

_{gx0}/μ (and v

_{gt0}/μ) ≥ 2 m/s. As previously discussed, the dispersion for the pulse-like and non-pulse-like ground motions is expected to be the same; thus, the values of \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\) for the pulse-like ground motions are also taken as 0.55. The aforementioned dispersion is a result of the random nature of the ground motion, which belongs to the aleatory uncertainty. Other sources of variability are referred to as the epistemic uncertainty, which is related to the lack of knowledge about the real structural properties and modeling approximations. Simultaneous consideration of both types of uncertainty involves an elaborate Monte Carlo simulation with appropriate distribution functions for the structural properties, which requires considerable effort. For simplicity, an approximate method based on the assumption that the effects of aleatory and epistemic sources are independent (FEMA, 2009) is adopted here. Assuming a recommended epistemic dispersion of 0.35 for average modeling quality, as suggested by FEMA P-58-1 (FEMA, 2018), the total dispersion of the normalized PSD is \(\sqrt {0.55^{2} + 0.35^{2} } = 0.65\).

_{sx0}/μ, u

_{sy0}/μ, and u

_{st0}/μ) at a given level of corresponding normalized PGV (v

_{gx0}/μ, v

_{gy0}/μ, and v

_{gt0}/μ) follows the lognormal distribution, the probability, P

_{f}, of exceeding the sliding displacement threshold, u

_{lim}, for given values of PGV = pgv and μ = μ

_{0}can be computed by

_{s0}represents the PSD of interest; \(\Phi\) is the standard normal cumulative distribution function; the median normalized PSD, (u

_{s0}/μ

_{0})

_{m}, is computed using Eq. (6.12) or Eq. (6.13); and the total dispersion, β

_{tot}, is taken as 0.65, as discussed previously. Figure 6.24 shows the fragility curves for some typical values of μ and u

_{lim}[Eqs. (6.12b) and (6.13a) were used in the computation], which clearly demonstrate the variation of P

_{f}as the PGV increases and the effects of the primary parameters. As shown in Fig. 6.24, for small values of u

_{lim}(e.g., u

_{lim}= 0.1 m) or large values of μ (e.g., μ = 0.4), there is no significant difference between the fragility curves of the non-pulse-like and pulse-like ground motions; as u

_{lim}increases or μ decreases, this difference becomes more significant and the SB structures subjected to pulse-like ground motions are more vulnerable in comparison with those subjected to non-pulse-like ground motions. This outcome agrees with the differences in the median normalized PSD and normalized PGV relationships depicted in Fig. 6.18 for both ground motion types.

## 6.5 Conclusions

_{sx0}/μ at a given level of v

_{gx0}/μ and u

_{st0}/μ at a given level of v

_{gt0}/μ can be modeled by the lognormal distribution. The relationship between (u

_{sx0}/μ)

_{m}and v

_{gx0}/μ and that between (u

_{st0}/μ)

_{m}and v

_{gt0}/μ are close to each other. The influence of T

_{x}, T

_{x}/T

_{y}, and α on (u

_{sx0}/μ)

_{m}and (u

_{st0}/μ)

_{m}is insignificant; thus, it is appropriate to conservatively use the maximum values of (u

_{sx0}/μ)

_{m}and (u

_{st0}/μ)

_{m}for the common ranges of T

_{x}, T

_{y}, and α in the design of SB structures. The lognormal standard deviation, \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) [and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\)], generally lies between 0.4 and 0.6 except for some cases when v

_{gx0}/μ (and v

_{gt0}/μ) is below 2.

_{sx0}/μ)

_{m}[and (u

_{st0}/μ)

_{m}] corresponding to the pulse-like records are close to those corresponding to the non-pulse-like records. When the normalized PGV exceeds a certain value (approximately 6–8 m/s), the value of (u

_{sx0}/μ)

_{m}[and (u

_{st0}/μ)

_{m}] for the pulse-like records starts to exceed the corresponding value for the non-pulse-like records, and the difference increases monotonically as v

_{gx0}/μ (and v

_{gt0}/μ) increases. The difference in the value of \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) [and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\)] for the two types of ground motions is small.