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Open Access 2023 | OriginalPaper | Buchkapitel

6. Peak Sliding Displacements of Sliding Base Structures Under Earthquake Excitation

verfasst von : Hong-Song Hu

Erschienen in: Sliding Base Structures

Verlag: Springer Nature Singapore

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Abstract

To ensure the safety of sliding base (SB) structures, the peak sliding displacement (PSD) should not exceed the predetermined sliding displacement threshold. Therefore, the PSD is an important response quantity that should be evaluated in design. In this chapter, the probability distribution of the PSD at a given level of ground motion intensity, and the influence of the vertical earthquake component and various structural and ground motion characteristics on the PSD are investigated. Simplified equations for estimating the PSD are also developed for design purposes.

6.1 Selection of Ground Motion Intensity Measure

The 180 ordinary ground motion records and 60 near-fault pulse-like records on site classes C and D, which has been presented in Chap. 4, will continue to be used in the response history analyses conducted in this chapter. In Chap. 3, some typical response histories of the sliding displacements are presented. The PSDs in the two principal directions (i.e., x and y directions), usx0 and usy0, 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, ust0, can also be determined by
$$ u_{st0} = \mathop {\max }\limits_{t} \sqrt {u_{sx} \left( t \right)^{2} + u_{sy} \left( t \right)^{2} } $$
(6.1)
In design, the PSDs of interest that are required to check the sliding displacement are reliant on the boundary shape of the sliding surface. For SB masonry structures (e.g., Nanda et al., 2015; Qamaruddin et al., 1986a), bond beams are generally constructed under the masonry walls as sliding elements, and the boundary shape of the sliding surface is generally rectangular to align with the building plane. For this case, it is necessary to check the sliding displacements separately in the two principal directions, and the PSDs required in design are usx0 and usy0. 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 ust0 with the sliding displacement threshold. The analyses that follow investigate all of usx0, usy0, and ust0, covering the two cases mentioned before.
To develop dependable earthquake excitation-based prediction models for the PSDs of SB structures, it is necessary to choose a suitable ground motion intensity measure (IM) that provides relatively small record-to-record variability of the PSDs at a given IM. The PGA and PGV, which rely exclusively on the ground motion characteristics, are the most traditional measures of ground motion intensity. Although there have been proven several IMs that consider both ground motion and structural properties [e.g., spectral acceleration at the first-mode period of the structure (Housner, 1941) and average spectral acceleration (Eads et al., 2015)] to be more efficient for seismic response assessment of fixed base (FB) structures, the changing dynamic property of a SB structure may render them unsuitable when sliding occurs. Thus, a feasible approach to assess the PSDs of SB structures involves choosing an IM that solely relies on the ground motion characteristics and analyzing the effect of distinct structural properties in isolation.
PGA and PGV are potential IMs due to their widespread usage among researchers and engineers, as well as the availability of corresponding attenuation relationships (Villaverde, 2009). To compare the efficiency (Luco & Cornell, 2007) of different IMs (i.e., their capability to produce small variability of the PSD at a given IM), response history analyses of a SB structure with Tx = Ty = 0.4 s (Tx = 2π/ωx and Ty = 2π/ωy), ξx = ξy = 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 usx0 and ust0 in relation to their respective PGAs (agx0 and agt0) and PGVs (vgx0 and vgt0), 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 ust0, Eqs. (6.2) and (6.3) are respectively used to compute the PGA and PGV corresponding to ust0, which are the maximum values of PGA and PGV over all the horizontal directions.
$$ a_{gt0} = \mathop {\max }\limits_{t} \sqrt {\ddot{u}_{gx} \left( t \right)^{2} + \ddot{u}_{gy} \left( t \right)^{2} } $$
(6.2)
$$ v_{gt0} = \mathop {\max }\limits_{t} \sqrt {\dot{u}_{gx} \left( t \right)^{2} + \dot{u}_{gy} \left( t \right)^{2} } $$
(6.3)
The superiority of PGV over PGA is evident from Fig. 6.1. Previous researchers (e.g., Jampole et al., 2020; Ryan & Chopra, 2004) have also observed this result. To further quantify the performance of different IMs, the results shown in Fig. 6.1 were also subjected to correlation coefficient computation; since a linear relationship may not be the best representation of the connection between the PSD and any of the IMs under consideration, the Spearman rank correlation coefficient (Maritz, 1995), ρs, for nonlinear correlations is used here. For μ = 0.1, the computed values of ρs are 0.38 (0.38) and 0.80 (0.85) for the correlations between usx0 and agx0 (ust0 and agt0), and usx0 and vgx0 (ust0 and vgt0), 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):
(1)
The initiation of sliding is dominated by the acceleration quantities; thus, initiating sliding through a pulse with larger PGV may not be easier as the larger PGV could be the result of longer duration instead of a larger acceleration amplitude.
 
(2)
Although the incremental velocity of a pulse can effectively describe the sliding excursion resulting from a velocity pulse, the incremental velocity of the largest pulse may not necessarily be in close proximity to the PGV of a ground motion record because the value of PGV is also affected by the initial conditions preceding the largest pulse.
 
(3)
The PSD obtained from seismic excitation of a SB structure is an accumulative result of multiple sliding excursions initiated by large velocity pulses, especially for lower friction levels. The efficiency of PGV is further decreased due to the accumulative effect, as it is typically associated with a dominant pulse.
 
Jampole et al. (2020) suggested a new IM, named EIGV, that is more effective in forecasting PSDs of rigid bodies exposed to earthquake ground motions. However, because the correlation between the PSD and PGV is acceptable and PGV is simple and well accepted by the engineering community as a ground motion IM, PGV is adopted herein.

6.2 Critical Parameters and Their Ranges

The parametric study presented in Chap. 5 indicates that the story number N generally does not influence the distribution of the equivalent lateral forces, which implies that the sliding displacement of a multistory SB structure should be close to that of the corresponding single-story SB structure with the same mass ratio and fundamental period. To verify this inference, the responses of a three-story and a single-story SB structure with the mass ratio α = 0.75 and the fundamental periods in the x and y directions equal to 0.4 s were computed. Figure 6.2 compares the probability distributions of the PSDs normalized by the friction coefficient μ of these two structures. As can be seen, the probability distributions of the normalized PSDs of the three-story and single-story structures almost coincide at a given level of the PGV normalized by μ. Therefore, single-story SB structures can be used to evaluate the PSDs of general SB structures.
As already presented in Chap. 4, the governing equations of single-story SB structures are as follows: For the stick phases,
$$ \left\{ \begin{aligned} & \ddot{u}_{rx} + 2\xi_{x} \omega_{x} \dot{u}_{rx} + \omega_{x}^{2} u_{rx} = - \ddot{u}_{gx} \\ & \ddot{u}_{ry} + 2\xi_{y} \omega_{y} \dot{u}_{ry} + \omega_{y}^{2} u_{ry} = - \ddot{u}_{gy} \\ \end{aligned} \right. $$
(6.4)
The precondition for the stick phases is
$$ \sqrt {\left( {\alpha \ddot{u}_{rx} + \ddot{u}_{gx} } \right)^{2} + \left( {\alpha \ddot{u}_{ry} + \ddot{u}_{gy} } \right)^{2} } < \left( {g + \ddot{u}_{gz} } \right)\mu_{s} $$
(6.5)
For the sliding phases,
$$ \left\{ {\begin{array}{*{20}l} {\ddot{u}_{sx} + \ddot{u}_{rx} + 2\xi_{x} \omega_{x} \dot{u}_{rx} + \omega_{x}^{2} u_{rx} = - \ddot{u}_{gx} } \hfill \\ {\ddot{u}_{sx} + \frac{{\dot{u}_{sx} }}{{\sqrt {\dot{u}_{sx}^{2} + \dot{u}_{sy}^{2} } }}\left( {g + \ddot{u}_{gz} } \right)\mu + \alpha \ddot{u}_{rx} = - \ddot{u}_{gx} } \hfill \\ {\ddot{u}_{sy} + \ddot{u}_{ry} + 2\xi_{y} \omega_{y} \dot{u}_{ry} + \omega_{y}^{2} u_{ry} = - \ddot{u}_{gy} } \hfill \\ {\ddot{u}_{sy} + \frac{{\dot{u}_{sy} }}{{\sqrt {\dot{u}_{sx}^{2} + \dot{u}_{sy}^{2} } }}\left( {g + \ddot{u}_{gz} } \right)\mu + \alpha \ddot{u}_{ry} = - \ddot{u}_{gy} } \hfill \\ \end{array} } \right. $$
(6.6)
As can be seen from Eqs. (6.4) to (6.6), the responses of SB structures are influenced by the friction coefficient (the dynamic and static friction coefficients are assumed to be the same), which is a critical parameter. By dividing both sides of Eqs. (6.4)–(6.6) by μ, it can be observed that the effects of μ can be incorporated into both the response quantities and ground motion IM, i.e., the normalized displacement quantities, \(\overline{u}_{rx} \left( t \right) = u_{rx} \left( t \right)/\mu\), \(\overline{u}_{ry} \left( t \right) = u_{ry} \left( t \right)/\mu\), \(\overline{u}_{sx} \left( t \right) = u_{sx} \left( t \right)/\mu\), and \(\overline{u}_{sy} \left( t \right) = u_{sy} \left( t \right)/\mu\), are dependent on the normalized IMs, vgx0/μ and vgy0/μ (where vgx0 and vgy0 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 (usx0/μ, usy0/μ, and ust0/μ) at various normalized PGVs (vgx0/μ, vgy0/μ, and vgt0/μ). Equations (6.4)–(6.6) shows that ωx, ωy, ξx, ξy, α, \(\ddot{u}_{gz} \left( t \right)\), and the horizontal ground motion waveform are other parameters that may affect the normalized PSDs.
The common ranges of the natural periods of the superstructure (Tx = 2π/ωx and Ty = 2π/ωy), the damping ratios (ξx and ξy), 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

The coupling of the friction forces in the two orthogonal directions [i.e., their resultant is equal to \(\left( {m + m_{b} } \right)\left( {g + \ddot{u}_{gz} } \right)\mu\)] causing the ground motion in one direction tends to decrease the friction force component in the orthogonal direction, resulting in an increase in the sliding displacement in that direction. However, this effect is reciprocal. Assuming using enough number of ground motions, the relationship between the average normalized PSD and normalized PGV in one direction that was obtained should be the same as that in the orthogonal direction under the circumstance of three-component seismic excitation. To confirm this inference, Fig. 6.3 displays the mean values of usx0/μ at each level of vgx0/μ considered and the data points, (vgy0/μ, usy0/μ), corresponding to the response in the y direction. These data were obtained from response history analyses of SB structures with Tx = Ty = 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 ≤ vgy0/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 usx0/μ and vgx0/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 usy0/μ and vgy0/μ agrees well with the curve of the mean usx0/μ versus vgx0/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

Figure 6.4 depicts the cumulative probability distribution of the normalized PSD (usx0/μ and ust0/μ) at four distinct levels of normalized PGV (vgx0/μ and vgt0/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 Tx = Ty = 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 usx0/μ at a given level of vgx0/μ and ust0/μ at a given level of vgt0/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
$$ \left( {u_{sx0} /\mu } \right)_{m} = \exp \left( {\sum\limits_{i = 1}^{n} {\ln \left( {u_{sx0} /\mu } \right)_{i} } /n} \right) $$
(6.7a)
$$ \left( {u_{st0} /\mu } \right)_{m} = \exp \left( {\sum\limits_{i = 1}^{n} {\ln \left( {u_{st0} /\mu } \right)_{i} } /n} \right) $$
(6.7b)
$$ \sigma_{{\ln \left( {u_{sx0} /\mu } \right)}} = \sqrt {\frac{{\sum\nolimits_{i = 1}^{n} {\left[ {\ln \left( {u_{sx0} /\mu } \right)_{i} - \ln \left( {u_{sx0} /\mu } \right)_{m} } \right]^{2} } }}{n - 1}} $$
(6.8a)
$$ \sigma_{{\ln \left( {u_{st0} /\mu } \right)}} = \sqrt {\frac{{\sum\nolimits_{i = 1}^{n} {\left[ {\ln \left( {u_{st0} /\mu } \right)_{i} - \ln \left( {u_{st0} /\mu } \right)_{m} } \right]^{2} } }}{n - 1}} $$
(6.8b)
where (usx0/μ)m and (ust0/μ)m are the medians of usx0/μ and ust0/μ, respectively; \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\) are the lognormal standard deviations of usx0/μ and ust0/μ, respectively; (usx0/μ)i and (ust0/μ)i are the observed value; and n is the sample size. However, for vgx0/μ (and vgt0/μ) ≤ 2 m/s, some observed values of usx0/μ (and ust0/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, (usx0/μ)m and (ust0/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.
$$ \sigma_{{\ln \left( {u_{sx0} /\mu } \right)}} = {{\ln}}\left[ {\frac{{\left( {u_{sx0} /\mu } \right)_{84\% } }}{{\left( {u_{sx0} /\mu } \right)_{50\% } }}} \right] $$
(6.9a)
$$ \sigma_{{\ln \left( {u_{st0} /\mu } \right)}} = {{\ln}}\left[ {\frac{{\left( {u_{st0} /\mu } \right)_{84\% } }}{{\left( {u_{st0} /\mu } \right)_{50\% } }}} \right] $$
(6.9b)
where (usx0/μ)50% and (usx0/μ)84% are the counted median and counted 84th percentile of usx0/μ, respectively; and (ust0/μ)50% and (ust0/μ)84% are the counted median and counted 84th percentile of ust0/μ, respectively.
Figure 6.4 also displays the fitted lognormal distribution functions for each of the four levels of vgx0/μ and vgt0/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 vgx0/μ (and vgt0/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 vgx0/μ (and vgt0/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 usx0/μ at a given level of vgx0/μ, and ust0/μ at a given level of vgt0/μ.

6.3.3 Effect of the Vertical Ground Motion Component

To investigate the influence of the vertical component on the PSD, the computation was carried out on the responses of SB structures exposed solely to the two horizontal components of the 180 non-pulse-like ground motions. Figure 6.5 shows a comparison between the normalized PSD probability distribution under the two-component excitation and that of corresponding three-component excitation. The normalized PSD probability distributions for both cases in this figure are almost the same at a given normalized PGV level, with only slight differences in the upper portions of the cumulative distribution curves. This result indicates that the effect of the vertical component on the PSD is negligible. Shao and Tung (1999) and Konstantinidis and Nikfar (2015) have also arrived at comparable conclusions regarding the sliding behavior of rigid bodies.

6.3.4 Effects of the Superstructure Natural Period and Mass Ratio

Figures 6.6 and 6.7 present the relationships between (usx0/μ)m and vgx0/μ, and (ust0/μ)m and vgt0/μ, respectively, for different values of Tx and α, which were determined by using the 180 non-pulse-like ground motion records and assuming Tx = Ty. As shown in these figures, the trend of (usx0/μ)m changing as vgx0/μ increases closely resembles the trend of (ust0/μ)m changing as vgt0/μ increases. In comparison with the normalized PGV, the influence of Tx and α on (usx0/μ)m and (ust0/μ)m is not so significant. To further investigate the combined effects of Tx and α on (usx0/μ)m and (ust0/μ)m, (usx0/μ)m and (ust0/μ)m are plotted against Tx 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 Tx = 0. As can be seen in Figs. 6.8 and 6.9, when Tx ≤ 0.4 s, the mass ratio basically has a negligible effect on (usx0/μ)m and (ust0/μ)m; when Tx > 0.4 s, the influence of α becomes slightly more significant, and (usx0/μ)m and (ust0/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 (usx0/μ)m or (ust0/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 (usx0/μ)m and (ust0/μ)m generally first increase and then decrease as Tx increases, and the differences between the maximum and minimum values of (usx0/μ)m and (ust0/μ)m for Tx 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 (usx0/μ)m and (ust0/μ)m for the range of Tx considered to conservatively estimate the PSDs of possible SB structures.
Figures 6.10 and 6.11 present the relationships between \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) and vgx0/μ, and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\) and vgt0/μ, respectively, for different values of Tx and α. For some cases when vgx0/μ (and vgt0/μ) = 1 m/s, the values of (usx0/μ)50% [and (ust0/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 vgx0/μ (and vgt0/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 vgx0/μ (and vgt0/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 Tx, 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 vgx0/μ (and vgt0/μ) = 1.5 and 2 m/s and even larger values are obtained when Tx = 1 s and α ≤ 0.7.
The combined effects of Tx 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 vgx0/μ (and vgt0/μ) ≤ 2 m/s. For any given level of vgx0/μ (and vgt0/μ), the maximum value of \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) [and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\)] is obtained at Tx = 1 s; this value is 0.87 (and 1.12) for vgx0/μ (and vgt0/μ) = 2 m/s and is around 0.63 (and 0.58) for all other levels of vgx0/μ (and vgt0/μ) ≥ 4 m/s. The value of Tx at which the minimum \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) [and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\)] is obtained varies for different levels of vgx0/μ (and vgt0/μ). For a given level of vgx0/μ (and vgt0/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 vgx0/μ (and vgt0/μ) = 2 m/s and ranges from 0.49 to 0.58 (and 0.45–0.55) for vgx0/μ (and vgt0/μ) ≥ 4 m/s.
The equality Tx = Ty is employed in all of the aforementioned analyses. To investigate the possible effect of Tx/Ty on the PSD, the values of (usx0/μ)m and (ust0/μ)m corresponding to different values of Tx/Ty 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 Tx/Ty has a negligible influence. Therefore, the results obtained for Tx = Ty can represent those of the other Tx/Ty within the range considered.

6.3.5 Effect of Near-Fault Pulses

Distinct pulses in near-fault ground motions affected by forward directivity may result in distinct sliding response characteristics as compared to ordinary ground motions. To investigate this effect, the (usx0/μ)m versus vgx0/μ and (ust0/μ)m versus vgt0/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 vgx0/μ ≤ 4 m/s (and vgt0/μ ≤ 6 m/s), the values of (usx0/μ)m [and (ust0/μ)m] corresponding to the pulse-like records are close to those corresponding to the non-pulse-like records. When vgx0/μ exceeds 6 m/s (and vgt0/μ exceeds 8 m/s), the value of (usx0/μ)m [and (ust0/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 vgx0/μ (and vgt0/μ) increases. To interpret the underlying reason for this phenomenon, Fig. 6.20 (Tx = Ty = 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 usx0/μ in each (non-pulse-like or pulse-like) group. By comparing the time histories presented in Fig. 6.20b for vgx0/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 (usx0/μ)m in comparison with the non-pulse-like ground motion. However, when vgx0/μ is small, as illustrated in Fig. 6.20a for vgx0/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.
$$ u_{s,\max } = \frac{{a_{p}^{2} T_{p}^{2} }}{4\mu g} - \frac{1}{2}a_{p} T_{p}^{2} + \frac{1}{4}T_{p}^{2} \mu g $$
(6.10)
where us,max is the PSD of the rigid block; and ap and Tp are the peak acceleration and duration of the half-sine pulse, respectively. Dividing both sides of Eq. (6.10) by μ and replacing apTp with πvgi/2 (where vpi is the peak velocity of the half-sine pulse), lead to
$$ \frac{{u_{s,\max } }}{\mu } = \frac{{\pi^{2} }}{16g}\left( {\frac{{v_{pi} }}{\mu }} \right)^{2} - \frac{\pi }{4}\frac{{v_{pi} }}{\mu }T_{p} + \frac{1}{4}T_{p}^{2} g $$
(6.11)
Equation (6.11) indicates that the quadratic relationship between normalized PSD and normalized PGV remains when a half-sine pulse is applied for excitation, that is, as the normalized PGV increases, the rate of increase of the normalized PSD with respect to it also increases, i.e., the effect of the prominent long-period velocity pulse is more significant when normalized PGV levels are higher.
Figures 6.21 and 6.22 compare the \(\sigma_{{{{\ln(u}}_{{{{sx0}}}} /\mu {)}}}\) versus vgx0/μ and \(\sigma_{{{{\ln(u}}_{{{{st0}}}} /\mu {)}}}\) versus vgt0/μ curves, respectively, of the pulse-like records with those of the non-pulse-like records. As shown in this figure, when vgx0/μ (and vgt0/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 vgx0/μ (and vgt0/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

From the investigations in the preceding section, we know that the influence of Tx and α on (usx0/μ)m [and (ust0/μ)m] is limited in comparison with that of vgx0/μ (and vgt0/μ). Therefore, in the design of SB structures, it is advisable to use the maximum values of (usx0/μ)m and (ust0/μ)m conservatively for the common range of Tx. The maximum (usx0/μ)m versus vgx0/μ and maximum (ust0/μ)m versus vgt0/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 (usx0/μ)m and vgx0/μ, as well as the maximum (ust0/μ)m and vgt0/μ, and the obtained regression formulae are as follows:
(1)
For the non-pulse-like ground motions,
$$ \left( {u_{sx0} /\mu } \right)_{m} = {0}{{.0052}}\left( {v_{gx0} /\mu } \right)^{2} + 0.261\left( {v_{gx0} /\mu } \right) - 0.254 \ge 0 $$
(6.12a)
$$ \left( {u_{st0} /\mu } \right)_{m} = {0}{{.0047}}\left( {v_{gt0} /\mu } \right)^{2} + 0.283\left( {v_{gt0} /\mu } \right) - 0.308 \ge 0 $$
(6.12b)
 
(2)
For the near-fault pulse-like ground motions,
$$ \left( {u_{sx0} /\mu } \right)_{m} = {0}{{.017}}\left( {v_{gx0} /\mu } \right)^{2} + 0.257\left( {v_{gx0} /\mu } \right) - 0.341 \ge 0 $$
(6.13a)
$$ \left( {u_{st0} /\mu } \right)_{m} = {0}{{.020}}\left( {v_{gt0} /\mu } \right)^{2} + 0.190\left( {v_{gt0} /\mu } \right) - 0.282 \ge 0 $$
(6.13b)
where (usx0/μ)m and (ust0/μ)m are in m, and vgx0/μ and vgt0/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, R2, 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.
 
It is worth mentioning here that Ryan and Chopra (2004) also proposed a design equation for calculating the median peak displacements, (ust0)m, of friction pendulum isolators:
$$ \left( {u_{st0} } \right)_{m} = \frac{4.36}{{4\pi^{2} }}T_{b}^{0.14} \eta^{{\left( { - 0.99 - 0.20\ln \eta } \right)}} \max \left( {v_{gx0} ,v_{gy0} } \right) $$
(6.14)
where Tb is the isolation period, and \(\eta\) is defined as
$$ \eta = \frac{\mu g}{{\omega_{d} \max \left( {v_{gx0} ,v_{gy0} } \right)}} $$
(6.15)
ωd 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) [ωd = 3.05 is adopted, as done by Ryan and Chopra (2004), and Tb 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.
As demonstrated in the preceding section, there does not exhibit a clear trend for the influence of the structural parameters, Tx, Tx/Ty, 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 vgx0/μ (and vgt0/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 vgx0/μ (and vgt0/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 \(\sqrt {0.55^{2} + 0.35^{2} } = 0.65\).
The fragility curve is an effective approach in assessing the seismic vulnerability of SB structures caused by excessive sliding, which presents the probabilities of exceeding a specified sliding displacement threshold at various levels of ground motion intensity. Since the normalized PSD (usx0/μ, usy0/μ, and ust0/μ) at a given level of corresponding normalized PGV (vgx0/μ, vgy0/μ, and vgt0/μ) follows the lognormal distribution, the probability, Pf, of exceeding the sliding displacement threshold, ulim, for given values of PGV = pgv and μ = μ0 can be computed by
$$ \begin{aligned} P_{f} & = P\left( {U_{s0} > u_{\lim } \left| {PGV = pgv,\mu = \mu_{0} } \right.} \right) \\ & = P\left[ {\left( {U_{s0} /\mu_{0} } \right) > \left( {u_{\lim } /\mu_{0} } \right)\left| {\left( {PGV/\mu } \right) = \left( {pgv/\mu_{0} } \right)} \right.} \right] \\ & = 1 - \Phi \left( {\frac{{\ln \left( {u_{\lim } /\mu_{0} } \right) - \ln \left( {u_{s0} /\mu_{0} } \right)_{m} }}{{\beta_{tot} }}} \right) \\ \end{aligned} $$
(6.16)
where Us0 represents the PSD of interest; \(\Phi\) is the standard normal cumulative distribution function; the median normalized PSD, (us0/μ0)m, is computed using Eq. (6.12) or Eq. (6.13); and the total dispersion, βtot, is taken as 0.65, as discussed previously. Figure 6.24 shows the fragility curves for some typical values of μ and ulim [Eqs. (6.12b) and (6.13a) were used in the computation], which clearly demonstrate the variation of Pf as the PGV increases and the effects of the primary parameters. As shown in Fig. 6.24, for small values of ulim (e.g., ulim = 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 ulim 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

This chapter presents a comprehensive study on the peak sliding displacements of SB structures subjected to three-component earthquake excitations. The PSDs in both the two main directions and with respect to the origin are taken into account. PGV is chosen as the ground motion IM because it exhibits a higher correlation with PSD compared to PGA and its attenuation relationship is conveniently accessible for design use.
It is possible for an individual ground motion to exhibit significant differences in its PSDs between the two orthogonal horizontal directions, despite both directions having the same PGVs; however, on average, the relationship between the normalized PSD and normalized PGV is essentially identical for the two orthogonal directions. The effect of the vertical ground motion component on the PSD is negligible. The probability distributions of usx0/μ at a given level of vgx0/μ and ust0/μ at a given level of vgt0/μ can be modeled by the lognormal distribution. The relationship between (usx0/μ)m and vgx0/μ and that between (ust0/μ)m and vgt0/μ are close to each other. The influence of Tx, Tx/Ty, and α on (usx0/μ)m and (ust0/μ)m is insignificant; thus, it is appropriate to conservatively use the maximum values of (usx0/μ)m and (ust0/μ)m for the common ranges of Tx, Ty, 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 vgx0/μ (and vgt0/μ) is below 2.
When the normalized PGV is small, the values of (usx0/μ)m [and (ust0/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 (usx0/μ)m [and (ust0/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 vgx0/μ (and vgt0/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.
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Metadaten
Titel
Peak Sliding Displacements of Sliding Base Structures Under Earthquake Excitation
verfasst von
Hong-Song Hu
Copyright-Jahr
2023
Verlag
Springer Nature Singapore
DOI
https://doi.org/10.1007/978-981-99-5107-9_6