Peak Superstructure Responses of Single-Story Sliding Base Structures Under Earthquake Excitation

Abstract

Compared to multistory sliding base (SB) structures, single-story SB structure is simple and suitable for acquiring the critical parameters influencing the structural response. For this reason, this chapter focuses on the peak superstructure responses of single-story SB structures subjected to three-component earthquake excitation. The influence of the vertical earthquake component and various structural and ground motion characteristics on the peak superstructure response are investigated, and simplified design equations are developed. The developed equations also lay a foundation for the design of multistory SB structures, which will be discussed in the next chapter.

4.1 Critical Parameters and Their Ranges

In Chap. 3, the equations of motion for multistory SB structures have been derived. For single-story SB structures (Fig. 4.1), the governing equations can be simplified to the following forms:

$$\left\{ \begin{aligned} & \ddot{u}_{rx} + 2\xi_{x} \omega_{x} \dot{u}_{rx} + \omega_{x}^{2} u_{rx} = - \ddot{u}_{gx} \\ & \ddot{u}_{ry} + 2\xi_{y} \omega_{y} \dot{u}_{ry} + \omega_{y}^{2} u_{ry} = - \ddot{u}_{gy} \\ & \sqrt {\left( {\alpha \ddot{u}_{rx} + \ddot{u}_{gx} } \right)^{2} + \left( {\alpha \ddot{u}_{ry} + \ddot{u}_{gy} } \right)^{2} } < \left( {g + \ddot{u}_{gz} } \right)\mu_{s} \\ \end{aligned} \right.$$

(4.1)

Fig. 4.1

Single-story SB structure: a schematic plot; and b simplified model

for the stick phases, and

$$\left\{ \begin{aligned} & \ddot{u}_{sx} + \ddot{u}_{rx} + 2\xi_{x} \omega_{x} \dot{u}_{rx} + \omega_{x}^{2} u_{rx} = - \ddot{u}_{gx} \\ & \ddot{u}_{sx} + \frac{{\dot{u}_{sx} }}{{\sqrt {\dot{u}_{sx}^{2} + \dot{u}_{sy}^{2} } }}\left( {g + \ddot{u}_{gz} } \right)\mu + \alpha \ddot{u}_{rx} = - \ddot{u}_{gx} \\ & \ddot{u}_{sy} + \ddot{u}_{ry} + 2\xi_{y} \omega_{y} \dot{u}_{ry} + \omega_{y}^{2} u_{ry} = - \ddot{u}_{gy} \\ & \ddot{u}_{sy} + \frac{{\dot{u}_{sy} }}{{\sqrt {\dot{u}_{sx}^{2} + \dot{u}_{sy}^{2} } }}\left( {g + \ddot{u}_{gz} } \right)\mu + \alpha \ddot{u}_{ry} = - \ddot{u}_{gy} \\ \end{aligned} \right.$$

(4.2)

for the sliding phases. In Eqs. (4.1) and (4.2), $\omega_{x} = \sqrt {{{k_{x} } / m}}$ and $\xi_{x} = c_{x} /\left( {2m\omega_{x} } \right)$ ($\omega_{y} = \sqrt {{{k_{y} } / m}}$ and $\xi_{y} = c_{y} /(2m\omega_{y} )$) are the natural frequency and damping ratio, respectively, of the corresponding fixed base (FB) structure in the x-direction (y-direction), and $\alpha = m/\left( {m + m_{b} } \right)$ is defined as the mass ratio.

As revealed by Eqs. (4.1) and (4.2), the response of an SB structure is greatly affected by the friction coefficient; thus, it is meaningful to relate the intensity of the ground motion and the acceleration response of the superstructure to the friction coefficient, μ. To consider the response in the x-direction, we introduce

$$u_{stx}^{*} = \frac{\mu g}{{\omega_{x}^{2} }}$$

(4.3)

which is the displacement of the corresponding FB structure when subjected to a static force, mgμ, in the x-direction. Dividing Eqs. (4.1) and (4.2) by $u_{stx}^{*}$ leads to the following equations:

$$\left\{ \begin{aligned} & \ddot{\overline{u}}_{rx} + 2\xi_{x} \omega_{x} \dot{\overline{u}}_{rx} + \omega_{x}^{2} \overline{u}_{rx} = - \omega_{x}^{2} \frac{{a_{gx0} }}{\mu g}\ddot{\overline{u}}_{gx} \\ & \ddot{\overline{u}}_{ry} + 2\xi_{y} \omega_{x} \left( {\frac{{\omega_{y} }}{{\omega_{x} }}} \right)\dot{\overline{u}}_{ry} + \omega_{x}^{2} \left( {\frac{{\omega_{y} }}{{\omega_{x} }}} \right)^{2} \overline{u}_{ry} = - \omega_{x}^{2} \frac{{a_{gx0} }}{\mu g}\frac{{a_{gy0} }}{{a_{gx0} }}\ddot{\overline{u}}_{gy} \\ & \sqrt {\left( {\alpha \ddot{\overline{u}}_{rx} + \frac{{a_{gx0} }}{\mu g}\ddot{\overline{u}}_{gx} \omega_{x}^{2} } \right)^{2} + \left( {\alpha \ddot{\overline{u}}_{ry} + \frac{{a_{gy0} }}{\mu g}\ddot{\overline{u}}_{gy} \omega_{x}^{2} } \right)^{2} } < \left( {1 + \frac{{\ddot{u}_{gz} }}{g}} \right)\omega_{x}^{2} \frac{{\mu_{s} }}{\mu } \\ \end{aligned} \right.$$

(4.4)

and

$$\left\{ \begin{aligned} & \ddot{\overline{u}}_{sx} + \ddot{\overline{u}}_{rx} + 2\xi_{x} \omega_{x} \dot{\overline{u}}_{rx} + \omega_{x}^{2} \overline{u}_{rx} = - \omega_{x}^{2} \frac{{a_{gx0} }}{\mu g}\ddot{\overline{u}}_{gx} \\ & \ddot{\overline{u}}_{sx} + \frac{{\dot{\overline{u}}_{sx} }}{{\sqrt {\dot{\overline{u}}_{sx}^{2} + \dot{\overline{u}}_{sy}^{2} } }}\left( {1 + \frac{{\ddot{u}_{gz} }}{g}} \right)\omega_{x}^{2} + \alpha \ddot{\overline{u}}_{rx} = - \omega_{x}^{2} \frac{{a_{gx0} }}{\mu g}\ddot{\overline{u}}_{gx} \\ & \ddot{\overline{u}}_{sy} + \ddot{\overline{u}}_{ry} + 2\xi_{y} \omega_{x} \left( {\frac{{\omega_{y} }}{{\omega_{x} }}} \right)\dot{\overline{u}}_{ry} + \omega_{x}^{2} \left( {\frac{{\omega_{y} }}{{\omega_{x} }}} \right)^{2} \overline{u}_{ry} = - \omega_{x}^{2} \frac{{a_{gx0} }}{\mu g}\frac{{a_{gy0} }}{{a_{gx0} }}\ddot{\overline{u}}_{gy} \\ & \ddot{\overline{u}}_{sy} + \frac{{\dot{\overline{u}}_{sy} }}{{\sqrt {\dot{\overline{u}}_{sx}^{2} + \dot{\overline{u}}_{sy}^{2} } }}\left( {1 + \frac{{\ddot{u}_{gz} }}{g}} \right)\omega_{x}^{2} \left( {\frac{{\omega_{y} }}{{\omega_{x} }}} \right)^{2} + \alpha \ddot{\overline{u}}_{ry} = - \omega_{x}^{2} \frac{{a_{gx0} }}{\mu g}\frac{{a_{gy0} }}{{a_{gx0} }}\ddot{\overline{u}}_{gy} \\ \end{aligned} \right.$$

(4.5)

where $\overline{u}_{rx} \left( t \right) = {{u_{rx} \left( t \right)} / {u_{stx}^{*} }}$ and $\overline{u}_{ry} \left( t \right) = {{u_{ry} \left( t \right)} / {u_{stx}^{*} }}$ are the normalized relative displacements and $\overline{u}_{sx} \left( t \right) = {{u_{sx} \left( t \right)} / {u_{stx}^{*} }}$ and $\overline{u}_{sy} \left( t \right) = {{u_{sy} \left( t \right)} / {u_{stx}^{*} }}$ are the normalized sliding displacements; $\dot{\overline{u}}_{rx} \left( t \right)$, $\dot{\overline{u}}_{ry} \left( t \right)$, $\dot{\overline{u}}_{sx} \left( t \right)$ and $\dot{\overline{u}}_{sy} \left( t \right)$ and $\ddot{\overline{u}}_{rx} \left( t \right)$, $\ddot{\overline{u}}_{ry} \left( t \right)$, $\ddot{\overline{u}}_{sx} \left( t \right)$ and $\ddot{\overline{u}}_{sy} \left( t \right)$ are the corresponding normalized velocities and accelerations, respectively; a_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.

The maximum earthquake force applied to the superstructure in the x-direction can be expressed as mA_x, where A_x is the peak pseudoacceleration (Chopra, 2001) in the x-direction, which is defined as

$$A_{x} = \omega_{x}^{2} \times \max \left( {\left| {u_{rx} (t)} \right|} \right)$$

(4.6)

By using Eq. (4.3), the normalized peak pseudoacceleration, A_x/μg, can be written as

$$\frac{{A_{x} }}{\mu g} = \frac{{\omega_{x}^{2} \times \max \left( {\left| {u_{rx} (t)} \right|} \right)}}{\mu g} = \frac{{\max \left( {\left| {u_{rx} (t)} \right|} \right)}}{{u_{stx}^{*} }} = \max \left( {\left| {\overline{u}_{rx} \left( t \right)} \right|} \right)$$

(4.7)

Therefore, A_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.

The first step towards conducting parametric studies in the following sections is to investigate the ranges of the critical parameters in accordance with common practice. Due to the fact that SB structures are designed for low-rise buildings, the natural periods of the superstructure, denoted as T_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

The ground motion records were selected for each site class defined by ASCE 7-10 (ASCE, 2010) from the Pacific Earthquake Engineering Research Center-Next Generation Attenuation (PEER-NGA) database. Since SB isolation takes effect primarily under significant ground accelerations, it is advisable to select ground motion records that have sufficiently large PGAs. by doing so, an extremely small value of μ can also be avoided as far as possible when conducting a parametric study. Therefore, for site classes C and D, only the ground motion records in which the peak value of the x-component, a_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.

Fig. 4.2

Distribution of magnitudes and source-to-site distances of the ground motion records: a site class B; b site class C; c site class D; and d site class E

The previous section mentioned that the response of the superstructure in the x-direction is influenced by both the y and z (vertical) components of the ground acceleration; therefore, comparing the PGAs of all three components is necessary to investigate the significance of the interaction between the two horizontal directions and the impact of the vertical component. Figure 4.3 shows the distributions of a_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.

Fig. 4.3

Distributions of a a_gy0/a_gx0; and b a_gz0/a_gx0

4.3 Normalized Ground Motion Intensity for the Initiation of Sliding

If the third equation of Eq. (4.1) is satisfied throughout the entire excitation history, sliding will not occur; thus, the critical static friction coefficient, $\mu_{cr}$, for the initiation of sliding can be determined using the following equation:

$$\mu_{cr} = \mathop {\max }\limits_{t} \left( {\frac{{\sqrt {\left( {\alpha \ddot{u}_{rx} \left( t \right) + \ddot{u}_{gx} \left( t \right)} \right)^{2} + \left( {\alpha \ddot{u}_{ry} \left( t \right) + \ddot{u}_{gy} \left( t \right)} \right)^{2} } }}{{g + \ddot{u}_{gz} \left( t \right)}}} \right)$$

(4.8)

where $\ddot{u}_{rx} \left( t \right)$ and $\ddot{u}_{ry} \left( t \right)$ are computed using the first and second equations of Eq. (4.1). According to Eq. (4.8), disregarding the influence of the vertical component, $\ddot{u}_{gz} \left( t \right)$, there exists a linear correlation between $\mu_{cr}$ and PGA; and hence, $\max \left( {a_{gx0} ,a_{gy0} } \right)/\mu g$ can be used as a more generalized indicator for determining the occurrence of sliding, i.e., sliding occurs when $\max \left( {a_{gx0} ,a_{gy0} } \right)/\mu_{s} g > \max \left( {a_{gx0} ,a_{gy0} } \right)/\mu_{cr} g$.

Figure 4.4 plots the counted median, counted 5th percentile, and counted 95th percentile values of $\max \left( {a_{gx0} ,a_{gy0} } \right)/\mu_{cr} g$ for the 120 selected ground motion records for site class D. In the computation, T_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.

Fig. 4.4

Typical statistical values of $\max \left( {a_{gx0} ,a_{gy0} } \right)/\mu_{cr} g$: a counted median; b counted 5th percentile; and c counted 95th percentile

4.4 Parametric Study for the Maximum Superstructure Response

According to the previous discussion on the range of a_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).

Figure 4.5 shows each individual value of $A_{x} /\mu g$ computed using the 90 non-pulse-like records for site class D in addition to the mean, the mean plus one standard deviation (SD) (corresponding to the 84th percentile value of the normal distribution), the counted median and the counted 84th percentile. In this computation, μ_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.

Fig. 4.5

Individual and some statistical values of $A_{x} /\mu g$: a $\alpha = 0.5$; and b $\alpha = 0.8$

Fig. 4.6

Comparison of probability densities of $A_{x} /\mu g$ corresponding to different groups: a $\alpha = 0.5$, $a_{gx0} /\mu g$ = 1; b $\alpha = 0.5$, $a_{gx0} /\mu g = 4$; c $\alpha$ = 0.8, $a_{gx0} /\mu g = 1$; and d $\alpha$ = 0.8, $a_{gx0} /\mu g = 4$

4.4.1 Comparison of the Response in Two Orthogonal Directions

Figure 4.7 shows the mean values of $A_{x} /\mu g$ at each level of $a_{gx0} /\mu g$, in addition to each individual value of $A_{y} /\mu g$ at the corresponding level of $a_{gy0} /\mu g$ computed using the 90 non-pulse-like records selected for site class D. In this computation, μ_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.

Fig. 4.7

Comparison of the relationship between the normalized peak pseudo-acceleration and the normalized PGA for the two orthogonal horizontal directions: a $\alpha$ = 0.5; and b $\alpha$ = 0.8

4.4.2 Effect of the Vertical Earthquake Component

The responses of SB structures under only the two horizontal components of earthquake excitation were also computed in order to study the effect of the vertical component. Figure 4.8 shows the ratios of A_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.

Fig. 4.8

Ratio of A_x under three-component excitation to that under the corresponding two-horizontal-component excitation: a $\alpha$ = 0.5, T_x = T_y = 0.3 s; b $\alpha$ = 0.5, a_gx0/μg = 4; c $\alpha$ = 0.8, T_x = T_y = 0.3 s; and d $\alpha$ = 0.8, a_gx0/$\mu$g = 4

4.4.3 Effect of the Natural Period of the Superstructure

Figure 4.9 shows the relationship between the mean A_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.

Fig. 4.9

Relationship between mean $A_{x} /\mu g$ and T_x: a $\alpha$ = 0.5; and b $\alpha$ = 0.8

In all the above analyses, T_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.

Fig. 4.10

Effect of T_x/T_y on the mean values of $A_{x} /\mu g$: a $\alpha$ = 0.5; and b $\alpha$ = 0.8

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

Based on the investigation mentioned above, $\mu$_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.

Fig. 4.11

Effect of μ_s/μ on the mean values of A_x/μg: a $\alpha$ = 0.5; and b $\alpha$ = 0.8

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

The mean values of A_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.

Fig. 4.12

Mean values of $A_{x} /\mu g$ for three magnitude intervals: a $\alpha$ = 0.5; and b $\alpha$ = 0.8

Figure 4.13 shows the mean values of $A_{x} /\mu g$ for three distance intervals. The influence of the source-to-site distance on the superstructure response, like that of the magnitude of the earthquake, is also insignificant.

Fig. 4.13

Mean values of $A_{x} /\mu g$ for three distance intervals: a $\alpha$ = 0.5; and b $\alpha$ = 0.8

4.4.6 Effect of Near-Fault Pulses

The acceleration, velocity, and displacement histories of near-fault ground motions influenced by forward directivity contain distinct pulses. To investigate the possible effects of these pulses, the mean values of $A_{x} /\mu g$ computed using the 30 pulse-like records and 30 non-pulse-like records both recorded for 0 < D < 14 km and site class D are compared in Fig. 4.14. When sliding basically does not occur, i.e., $a_{gx0} /\mu g$ = 0.25, the mean value of $A_{x} /\mu g$ corresponding to the non-pulse-like records is 1.16 times that corresponding to the pulse-like records. The corresponding FB structure has a larger response amplification factor, A_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.

Fig. 4.14

Effect of near-fault pulses on the mean values of $A_{x} /\mu g$: a $\alpha$ = 0.5; and b $\alpha$ = 0.8

4.4.7 Statistical Results for Each Site Class

Figure 4.15 shows the mean values of $A_{x} /\mu g$ for each site class. Herein, only the non-pulse-like records were used for the computation to maintain consistency. Thus, the numbers of ground motion records used for site classes B, C, D and E are 40, 90, 90 and 40, respectively. As shown in Fig. 4.15, except for $a_{gx0} /\mu g$ = 0.25 (which is basically equivalent to the FB case), as the mass ratio, $\alpha$, increases, the superstructure response reduces. This phenomenon can be easily explained by using the governing equations for the sliding phases during unidirectional excitation. Under unidirectional excitation in the x-direction, Eq. (4.2) can be simplified into

$$\ddot{u}_{rx} + 2\frac{{\xi_{x} }}{{\sqrt {1 - \alpha } }}\frac{{\omega_{x} }}{{\sqrt {1 - \alpha } }}\dot{u}_{rx} + \frac{{\omega_{x}^{2} }}{1 - \alpha }u_{rx} = \frac{{\dot{u}_{sx} }}{{\left| {\dot{u}_{sx} } \right|}}\frac{\mu g}{{1 - \alpha }}$$

(4.9)

Fig. 4.15

Mean values of $A_{x} /\mu g$ for each site class: a site class B; b site class C; c site class D; and d site class E

which is the differential equation of a single-degree-of-freedom (SDOF) system with a natural frequency of ${{\omega_{x} } / {\sqrt {1 - \alpha } }}$ and a damping ratio of ${{\xi_{x} } / {\sqrt {1 - \alpha } }}$ subjected to a step force corresponding to a static displacement of ${{\mu g} / {\omega_{x}^{2} }}$. The damping ratio of this equivalent system increases as $\alpha$ increases, leading to a general decrease in the response of u_rx as $\alpha$ increases.

After $a_{gx0} /\mu g$ exceeds 0.5, the mean $A_{x} /\mu g$ versus a_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.

Figure 4.16 presents the ratios of the mean value of $A_{x} /\mu g$ for site class C (D or E) to that for site class B, for $\alpha$ = 0.5 and 0.8, in order to investigate the effects of local site conditions on the superstructure response. It is evident that the response of the superstructure is affected by the local site conditions; as the site soil becomes softer, the mean values of A_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.

Fig. 4.16

Effect of local site conditions on the mean values of $A_{x} /\mu g$: a $\alpha$ = 0.5; and b $\alpha$ = 0.8

Figure 4.17 shows the coefficients of variation (COVs) of $A_{x} /\mu g$ for every site class in order to investigate the dispersion of the superstructure response at a specified value of $a_{gx0} /\mu g$. The COVs of the A_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.

Fig. 4.17

Coefficients of variation of $A_{x} /\mu g$ for each site class: a site class B; b site class C; c site class D; and d site class E

4.5 Simplified Equations for Estimating the Maximum Superstructure Response

For the design of SB structures, it is desirable to employ simplified equations to estimate the peak pseudoacceleration of the superstructure. Based on the preceding discussions, we know the following: (1) the relationship between $A_{x} /\mu g$ and $a_{gx0} /\mu g$ is basically identical to that between $A_{y} /\mu g$ and $a_{gy0} /\mu g$; (2) the dependencies of the mean value of $A_{x} /\mu g$ (or $A_{y} /\mu g$) on the vertical earthquake component, T_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 $A_{y} /\mu g$:

$$\frac{{A_{x} }}{\mu g} = \frac{{\beta_{1} \left( {a_{gx0} /\mu g} \right)^{{\beta_{2} }} }}{{\left( {a_{gx0} /\mu g} \right)^{{\beta_{2} }} + \beta_{3} }}\;\;{\text{and}}\;\;\frac{{A_{y} }}{\mu g} = \frac{{\beta_{1} \left( {a_{gy0} /\mu g} \right)^{{\beta_{2} }} }}{{\left( {a_{gy0} /\mu g} \right)^{{\beta_{2} }} + \beta_{3} }}$$

(4.10)

where $\beta_{1} ,\beta_{2} \,{\text{and}}\,\beta_{3}$ are the regression coefficients that depend on the site class and the mass ratio, $\alpha$. Equation (4.10) captures the trend of $A_{x} /\mu g$ $\left( {A_{y} /\mu g} \right)$ with respect to $a_{gx0} /\mu g$ $\left( {a_{gy0} /\mu g} \right)$, i.e., $A_{x} /\mu g$ → 0 when $a_{gx0} /\mu g$ → 0, and $A_{x} /\mu g$ approaches an upper limit when $a_{gx0} /\mu g$ → $+ \infty$. The Curve Fitting Toolbox of MATLAB (2014) was used to conduct nonlinear regression analyses for determining the values of the regression coefficients in Eq. (4.10). In order to assess conservativeness, regression analyses were conducted using results obtained from non-pulse-like records, considering that responses under pulse-like ground motions generally exhibit smaller values compared to those under non-pulse-like ground motions. Table 4.1 presents the values of $\beta_{1} ,\beta_{2} \,{\text{and}}\,\beta_{3}$ obtained for each site class and various values of $\alpha$. The values predicted through Eq. (4.10) are also compared to the values computed from response history analyses shown in Fig. 4.15. The mean values of $A_{x} /\mu g$ can be accurately estimated by the proposed equation. The values of $\beta_{1} ,\beta_{2} \,{\text{and}}\,\beta_{3}$ for the $\alpha$ values not included in Table 4.1 can be found by linear interpolation of the values from Table 4.1, because each regression coefficient has an approximately linear relationship with $\alpha$; if $\alpha$ > 0.9, using the $\beta_{1} ,\beta_{2} \,{\text{and}}\,\beta_{3}$ values for $\alpha$ = 0.9 can give conservative results.

Table 4.1 Values of the regression coefficients in Eq. (4.10)

As shown in the previous section, a normal distribution is appropriate for modeling the probability distribution of A_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

$$\delta_{{A_{x} /\mu g}} = \gamma_{1} \exp \left( { - \gamma_{2} \left( {a_{gx0} /\mu g} \right)} \right) + \gamma_{3} \;\;{\text{and}}\;\;\delta_{{A_{y} /\mu g}} = \gamma_{1} \exp \left( { - \gamma_{2} \left( {a_{gy0} /\mu g} \right)} \right) + \gamma_{3}$$

(4.11)

where $\delta_{{A_{x} /\mu g}} \,\left( {\delta_{{A_{y} /\mu g}} } \right)$ is the COV of $A_{x} /\mu g$ $\left( {A_{y} /\mu g} \right)$, and $\gamma_{1} ,\gamma_{2} \,{\text{and}}\,\gamma_{3}$ are the regression coefficients that depend on $\alpha$. Table 4.2 presents the values of $\gamma_{1} ,\gamma_{2} \,{\text{and}}\,\gamma_{3}$ based on the nonlinear regression analyses for various $\alpha$ values. Figure 4.18 shows a comparison of the values predicted using Eq. (4.11) and the values obtained through response history analyses. A satisfactory estimation for the COVs of $A_{x} /\mu g$ is yielded by the proposed equation. Similar to Eq. (4.10), for the values of $\alpha$ not listed in Table 4.2, the values of $\gamma_{1} ,\gamma_{2} {\text{ and }}\gamma_{3}$ can be calculated from the linear interpolation of those provided in Table 4.2; for $\alpha$ > 0.9, the values of $\gamma_{1} ,\gamma_{2} \,{\text{and}}\,\gamma_{3}$ for $\alpha = 0.9$ can be used.

Table 4.2 Values of the regression coefficients in Eq. (4.11)

Fig. 4.18

Comparison between the COVs of $A_{x} /\mu g$ computed using Eq. (4.11) and those computed from response history analyses

4.6 Conclusions

In this chapter, a comprehensive parametric investigation of the normalized peak pseudoacceleration of single-story SB structures subjected to three-component earthquake excitation is presented. The relationship between the normalized peak pseudoacceleration and the normalized PGA is basically identical for the two orthogonal horizontal directions. The horizontal response of the superstructure can be either reduced or increased by the vertical component of ground motion. If T_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.

The effects of the earthquake magnitude and the source-to-site distance are very small and can be neglected in practice. Superstructures typically exhibit smaller responses when subjected to pulse-like ground motions compared to non-pulse-like ones. Local site conditions have an effect on the response of the superstructure. For sites located on softer soil, a larger response is obtained, and the dependence on the local site conditions decreases as the normalized PGA increases. The trend of the COVs of A_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.

The mean values and COVs of A_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.