Seismic Overturning Fragility Analysis for Rigid Blocks Subjected to Floor Motions

Abstract

This paper investigates the seismic rocking-overturning fragility of freestanding rigid blocks subjected to one-sine acceleration pulses from a probabilistic perspective. An equivalent single-degree-of-freedom (SDOF) model with a bespoke discrete damper is used to simulate the responses of four blocks with varying geometries under excitation with various characteristics. The simulation results are used to perform an overturning fragility analysis and evaluate the performance of various intensity measures (IMs). An IM strip, referred to as a hybrid strip, can be observed in the analysis, within which both safe rocking and overturning occur. For IM values outside of the hybrid strip, there exists a clear distinction between these two states. In this study, we introduce the hybrid ratio, a parameter that can estimate the size of the hybrid strip of different IMs. The hybrid ratio is defined as the combination of two ratios of hybrid strip width and the two IM strip widths corresponding to safe rocking and overturning, respectively. The effect of the different analysis strip widths is also examined in the overturning fragility analysis. The results suggest that the IM determined by excitation magnitude, frequency, and block geometry parameters demonstrates its superiority compared with some well-known IMs by having the smallest hybrid ratio and coefficient of variation, as well as good robustness of the overturning fragility curves against the change of the analysis strip width.

1. Introduction

Building contents damage, an important part of earthquake-induced damage, has been attracting more and more interest from investigators since it was found that severe contents damage usually causes more significant financial loss compared with structural damage [1]. In the field of earthquake engineering, unanchored building contents are typically idealized as freestanding rigid blocks. During earthquake events, the motion of rigid blocks may be dominated by complex modes (e.g., sliding, twisting, rocking). During these complex motions, the blocks may impact each other or neighboring walls, and may even overturn. Among the dominating modes, rocking and sliding have attracted the most concern from investigators. Shenton [2] proposed an approach to distinguish different dominating modes of unanchored rigid blocks. The research addressing the sliding response can be found in the literature [3,4,5,6,7,8,9,10]. This study focuses on the rocking-overturning response of the freestanding rigid block, characterized by a partial bottom uplift and changing rotation center. The block can overturn when it undergoes a large enough rocking rotation.

As a pioneering work, the seminal framework proposed by Housner [11] for the seismic response evaluation of freestanding rigid blocks has been followed by many investigators, including experimental [12,13,14,15,16,17,18,19,20] and numerical efforts [21,22,23,24,25,26,27,28,29,30,31,32,33]. To predict the occurrence of overturning, Ishiyama [34] proposed overturning criteria involving the overturning acceleration and velocity, which are the minimum peak acceleration and velocity of the input excitation needed to overturn a rigid block, respectively. Kaneko and Hayashi [35] rearranged Ishiyama’s equation [34] to obtain the overturning acceleration considering the frequency of the input excitation. Then, Kaneko and Hayashi [36] proposed an equation to determine the relationship between the overturning ratio and the input excitation. Kuo et al. [37] proposed seismic evaluation criteria for the clutter response of medicine shelves via shake table tests. Similarly, Zhang and Makris [38] analyzed the condition of overturning and derived an overturning acceleration spectrum, which has been widely used in rocking-overturning research [39,40]. However, the above deterministic views and methods to predict overturning seem to be inconsistent with reality, as the rocking motion is an extremely complex dynamic process, and the rocking response is barely “nonrepeatable” [41,42,43,44]. Even minor adjustments of excitation or block geometry parameters may lead to significant differences of the response. This can be primarily attributed to the negative stiffness exhibited by the blocks [45] and the complex variability in energy dissipation [46]. Following the probabilistic point of view [42], fragility analysis, a commonly utilized approach in earthquake engineering [47], has been employed by numerous investigators [48,49,50] to evaluate the seismic performance of rigid blocks. This approach involves assessing a conditional probability for a damage measure exceeding a certain capacity limit state, given an IM value [51,52,53].

For assessing the rocking-overturning fragility, an incremental dynamic analysis (IDA)-based assessment method for the blocks subjected to floor motions is proposed by Liu et al. [54]. Nevertheless, rocking IDA curves generally differ from those of structural systems due to the frequent appearance of resurrections, the highly weaving non-monotonic behavior and the overall high variability [55], which presents new challenges to the application of IDA-pertinent approaches. Besides the IDA-based method, the maximum likelihood estimation (MLE) approach has been used to calculate the fragility function parameters for which the assumed statistical distribution attains the highest likelihood of producing the observed data [56]. Correspondingly, for a rocking-overturning fragility analysis, various IMs, including univariate IMs [57,58] and bivariate ones [56], have been proposed. A novel dimensionless IM, determined by the excitation magnitude, frequency, and block geometry parameters, was presented by Liu et al. [59] for evaluating the seismic rocking fragility, which demonstrates a closer correlation with the maximum rocking angle, thus achieve less dispersion of the fragility functions.

This paper takes a probabilistic insight into the rocking-overturning responses and evaluates the ability of a suite of IMs to describe the rocking-overturning response. The likelihood of overturning due to rocking is expressed with a ‘categorical’ response variable. In particular, a zero-valued (0) or one-valued (1) parameter suffices to describe overturning, because the rocking block either overturns or not [56]. The simulation of the rocking blocks is conducted using a reliable numerical model [60]. There exists a hybrid IM strip within which both overturning and safe rocking occur. For IM values outside of the hybrid strip, there exists a clear distinction between these two states. From the point of view of the hybrid strip, the hybrid ratio is proposed in this paper to quantitively compare the performance of various IMs in distinguishing the two states of safe rocking and overturning. Simulated overturning probabilities, the percentages of overturning occurrence within the specific IM value (strip), are used to generate fragility curves. In addition, the least squares method is used to obtain the parameters of the fragility functions. A novel IM [59], used for the first time in an overturning fragility analysis, receives the smallest hybrid ratio and the smallest coefficient of variation compared with some well-known IMs. Finally, the effect of different analysis strip widths is also examined, and the results show that the overturning fragility curves have good robustness against the change of the analysis strip width.

The following is an outline of the remaining sections. Section 2 provides the equation of motion for the rocking block and presents a discretely damped SDOF model, which is utilized to solve the rocking response. The experimental results are utilized to validate the numerical model. Utilizing an overturning acceleration spectrum [38], Section 3 demonstrates the rocking-overturning responses. In Section 4, an overturning fragility analysis is conducted in terms of a suite of IMs. A novel IM first used in predicting overturning successfully demonstrates its superiority by comparing it with some well-known ones. The final section provides some concluding remarks. Appendix A contains the notations of the variables utilized in the analysis.

2. Seismic Response of Rigid Blocks

2.1. Numerical Modeling

Consider a homogenous freestanding rectangular rigid block (Figure 1) that has the dimensions 2b × 2h, mass m. The center-of-mass (CM) of such a block coincides with its center-of-geometric. The block’s geometry can be fully described using two parameters: the size parameter $R=\sqrt{b^2+h^2}$ and the slenderness parameter, $\alpha =\mathrm{atan}\left(b/h\right)$. Assume that the coefficient of friction between the block and its rigid base is big enough so that rocking (including overturning when the rocking angle is too large) is the only dominating mode. Its rotational moment of inertia to the pivot point O or O’ is $I_{\mathrm{O}}=\frac{4}{3}m{R}^2$. In the simplest case, this system is SDOF.

Assuming that there is no jump and, as a result, the rigid block remains at the same position at the instant of impact, the equation of motion of an undamped freestanding block in Figure 1 can be expressed by:

$$I_{\mathrm{O}}\ddot{\theta }-m{\ddot{u}}_{0}H\left(\theta \right)+mgB\left(\theta \right)=0$$

(1)

where θ is the rotation angle, ${\ddot{u}}_0$ is the horizontal excitation and g is the gravity acceleration. The line connecting the current pivot point and the CM can be decomposed into vertical and horizontal components, i.e., H(θ) and B(θ), respectively (Equations (2) and (3)):

$$H\left(\theta \right)=R·\cos \left[\alpha ·\mathrm{sgn}\left(\theta \right)-\theta \right]$$

(2)

$$B\left(\theta \right)=R·\sin \left[\alpha ·\mathrm{sgn}\left(\theta \right)-\theta \right]$$

(3)

where sgn() is the sign function.

There is a threshold for the horizontal excitation acceleration ${\ddot{u}}_0$ to cause the block to start rocking (Equation (4)). The restoring moment M can be uniquely determined by θ in Equation (5).

$${\ddot{u}}_0\ge gb/h=g·\tan \alpha$$

(4)

$$M=mgR\sin \left[\alpha ·\mathrm{sgn}\left(\theta \right)-\theta \right]$$

(5)

To simulate the rocking responses, we adopt an equivalent lumped mass SDOF model [61]. Figure 2 displays the equivalent representation of a rocking block (Figure 2a) as an SDOF oscillator (Figure 2b). The equation of motion for such an oscillator subjected to a floor motion ${\ddot{u}}_0$ is given by,

$$I_O\ddot{\theta }+f_d\left(\dot{\theta }\right)+k\left(\theta \right)·\theta =-I_O\frac{{\ddot{u}}_0\cos \alpha }{R}$$

(6)

where $I_{\mathrm{O}}=\frac{4}{3}m{R}^2$ is the moment of inertia, $f_{\mathrm{d}}$ is the damping force and k is the tangent stiffness of the rotational spring to model the restoring moment of the rocking block. $I_{\mathrm{CM}}=\frac{m{R}^2}{3}$ is the additional moment of inertia to achieve the total moment of inertia $I_O$. To approximate the energy dissipation during rocking, a discrete damping force $f_{\mathrm{d}}\left(\dot{\theta }\right)$ proposed by Liu et al. [60] is implemented in this SDOF model. In this discrete damping model the viscous damping force $f_{\mathrm{d}}=c_{\mathrm{D}}\dot{\theta }$ has a limited application range of ±δα around the original position θ = 0. Beyond this range, the damping force is set to 0. Therefore, the energy dissipation during rocking occurs solely when the system traverses its original position (Equation (7)):

$$f_d\left(\theta ,\dot{\theta }\right)=\{\begin{array}{cc}{c}_D\dot{\theta },& \left|\theta \right|\le \delta \alpha \\ 0,& \left|\theta \right|>\delta \alpha \end{array}$$

(7)

where $c_{\mathrm{D}}$ is a nonconstant discrete viscous damping coefficient proportional to the angular velocity $\dot{\theta }$ before each impact (Figure 3) expressed by Equation (8).

$$c_D=\frac{I_O}{2\delta \alpha }\left(1-e\right)\left|{\dot{\theta }}_1\right|$$

(8)

where e is the restitution coefficient of the rocking block, which is defined as the ratio of the angular velocity after the impact to that before the impact floor [62], i.e., $e={\dot{\theta }}_2/{\dot{\theta }}_1$, with ${\dot{\theta }}_1$ and ${\dot{\theta }}_2$ representing the angular velocity of the rocking block before and after the impact. Under Housner’s assumptions [11], the rigid-body restitution coefficient e_R can be derived by the slenderness parameter α (Equation (9)). It is worth mentioning that the restitution coefficient e in the real world is usually smaller than e_R because of the localized nonlinearity of the colliding materials. To better match the real-world energy dissipation, the restitution coefficient in this study is assumed to be e = 0.95e_R. The numerical simulation was performed in OpenSees (Version 3.0.0) [63], a world-renowned finite element modeling platform for earthquake engineering. Further implemental details can be found in our former paper [60].

$$e_R=1-1.5{\sin }^2\alpha$$

(9)

2.2. Experimental Verifications

The performance of the discrete damping model has been validated by a comparison with the experimental results of Nasi [64,65,66]. Figure 4 displays six selected runs of rotation-time history derived from the experiment and the simulation. The comparison shows that this numerical model with discrete damping can correctly approximate the maximum rocking angle and predict overturning [59,60].

3. Overturning Acceleration Spectrum

The response results of the rigid rocking blocks are usually studied in terms of the rocking spectrum. Zhang and Makris [38] presented a two-dimensional overturning acceleration spectrum, with two axes of ω_P/P, and PFA/gtanα. Here, ω_P is the circular frequency of the one-sine pulse acceleration excitation, $P=\sqrt{3g/4R}$ is a block frequency parameter proposed by Housner [11] and PFA is the peak floor acceleration. In this study, the overturning acceleration spectrum proposed by Zhang and Makris [38] is simplified by only dividing into the overturning and safe rocking zone. For the subsequent overturning fragility analysis, a set of data were generated based on the overturning acceleration spectrum. We selected four models of sizes of typical furniture or equipment (Figure 5).

By exposing each model to 100 one-sine pulse excitations of different PFA and ω_P, we derived simulated results for 400 uniformly distributed cases (Figure 6a). A boundary can be observed between the overturning and safe area (Figure 6b), as Zhang and Makris [38] concluded. The occurrence of overturning corresponds to ω_P/P and PFA/gtanα, both widely-used IMs. However, using either one of these methods alone is insufficient for an overturning fragility analysis, highlighting the advantage of bivariate IMs [56]. Furthermore, this fact also consists of the motivation of this paper, which is that a good IM for an overturning fragility analysis should include as much of the excitation characteristics (e.g., PFA and ω_P) and the block geometry information (e.g., R and α), as mentioned in our former research [59].

It is worth noting that this study utilizes simple one-sine pulses as external excitations. Unlike rocking structures, building contents are subject to floor motion rather than ground motion. Previous work by D’Angela et al. [67] suggests that the input for freestanding bodies should better fit floor motion, and floor motions usually exhibit a relatively reduced record-to-record uncertainty. During earthquake events, the floor motion of a building is filtered by the structural system, resulting in dominant frequency components [68,69] (primarily related to the first natural frequency, or the first three natural frequencies). Although simplified motions may not fully capture the complexities of realistic floor motions under earthquake excitations, they do offer a foundation for understanding the response of rigid blocks and serve as a starting point for more sophisticated analyses.

4. Overturning Fragility Analysis

To determine the overturning fragility, one should first estimate the conditional probability P_f for the damage measure (DM) surpassing a pre-defined capacity limit state (LS), given an IM value:

$$P_{\mathrm{f}}=P\left(DM>LS|IM\right)$$

(10)

To facilitates the calculation of conditional probability P_f, Dimitrakopoulos and Paraskeva [56] have presented a probability tree diagram that takes into account the peculiarities of the rocking responses (Figure 7). The probability P_f is calculated by combining two likelihoods, i.e., the rocking-overturning probability (P_ro) and the probability of DM surpassing LS without overturning (P_ex). As the probability P_ex during safe rocking has been studied [59], this paper focuses on deriving the overturning probability P_ro, and a comparative study of a suite of IMs has been conducted on the performance of deriving the overturning probability.

4.1. Damage Measure and Limit States

In the real world, it is obvious that overturning occurs when the absolute peak rocking rotation |θ_max| reaches π/2. In the subsequent overturning fragility analysis, |θ_max| is used as the DM in this paper and LS = π/2 indicates the occurrence of overturning (Equation (11)).

$$DM=\left|{\theta }_{\max }\right|, LS=\pi /2$$

(11)

The dimensionless DM, normalized absolute peak rocking angle |θ_max|/α, has been widely adopted for rocking fragility analysis because it provides a straight insight into the degree of rocking response [56,57,58,59]. Its physical meaning is clear: a larger-than-zero value implies the existence of rocking, whereas higher values indicate more severe rocking and even overturning. This index (|θ_max|/α) is also used as a criteria of overturning, but there are different opinions on the threshold value. Some investigators use larger-than-one values to denote overturning [54,58,70], while some other researchers have suggested the possibility of the block surviving overturning and returning to its original configuration eventually with |θ_max|/α > 1.0 [38,71], which differs from the common quasi-static viewpoint of the seismic response. When overturning occurs the numerical response may tend to infinite values. Thus, Dimitrakopoulos and Paraskeva [56] pointed out that the threshold of overturning does not correspond to a particular rotation value (e.g., |θ_max|/α > 1.0 or even 1.5), but rather at infinite values in numerical simulations.

4.2. Intensity Measures

Liu et al. [59] evaluated eight frequently used dimensionless IMs in a rocking fragility analysis and proposed a novel dimensionless IM determined by the excitation magnitude, frequency and block geometry parameters. The novel IM shows obvious superiority in predicting the peak rocking rotation angle during safe rocking compared with eight well-known IMs. In this paper, we use the same IM suite to evaluate their performance in an overturning analysis. The detailed information on the IM suite is listed in

Table 1. Detailed information on the IM suite.

Intensity Measure	Expression	Physical Meaning	Ref
IM1	$\frac{{\omega }_P}{P}$	Dimensionless excitation frequency	[38]
IM2	$\frac{PFA}{g\tan \alpha }$	Dimensionless PFA	[38]
IM3	$\frac{P·PFV}{g\tan \alpha }$	Dimensionless PFV	[56]
IM4	$1.484{\left(\frac{PFA}{g\tan \alpha }\right)}^{1.644}{\left(\frac{{\omega }_P}{P}\right)}^{-2.013}$	Dimensionless combinations of PFA and floor motion frequency	[56]
IM5	$0.063{\left(\frac{PFA}{g\tan \alpha }\right)}^{2.954}{\left(\frac{{\omega }_P}{P}\right)}^{-0.942}$		[56]
IM6	${\left(\frac{PFA}{g\tan \alpha }\right)}^{0.52}{\left(\frac{{\omega }_P}{P}\right)}^{-0.48}$		[56]
IM7	${\left(\frac{PFA}{g\tan \alpha }\right)}^{0.6}{\left(\frac{{\omega }_P}{P}\right)}^{-0.4}$		[56]
IM8	$\frac{e_R{}^4·P·PFV}{g\tan \alpha }$	Dimensionless PFV considering the restitution coefficient	[58]
IM9	$\frac{PFA·T_P{}^2}{R\tan \alpha }$ *	Dimensionless floor displacement	[59]

* $T_P$ is the period of the one-sine pulse excitation.

and can also be found in [59].

Within this IM suite, IM₁ to IM₃ are univariate IMs with the physical meaning of dimensionless excitation frequency, dimensionless PFA and dimensionless PFV, respectively. IM₄ to IM₇ are bivariate IMs, which may increase the computational cost. IM₈ is an IM proposed by Sieber et al. [58] with the physical meaning of the dimensionless peak velocity considering the restitution coefficient e_R. The novel IM₉ [59], with the physical meaning of dimensionless displacement, explicitly considers the excitation (PFA and ω_P) and the block geometric (R and α).

4.3. Hybrid Strip and Hybrid Ratio

Figure 8 displays the simulated rocking responses against the varying values of different IMs. There exists a particular hybrid IM strip (HS), within which both safe rocking and overturning occur, and outside of which a clear distinction can be made between the two states of safe rocking and overturning. To quantitively estimate the size of the HS, the hybrid ratio (HR) is presented in this paper to compare the performance of various IMs for predicting overturning (Figure 8). The parameter HR defined in Equation (12) can assess the performance of different IMs to evaluate whether a rigid block will overturn if subjected to an excitation.

$$HR=\sqrt{{\left(\frac{W_{HS}}{R_S}\right)}^2+{\left(\frac{W_{HS}}{R_O}\right)}^2}$$

(12)

where W_HS is the width of the hybrid strip, R_S and R_O are the range of IM corresponding to safe rocking and the range of IM corresponding to overturning, respectively. Generally, a smaller HR for a type of IM indicates a smaller size of the hybrid strip relative to the overall IM range, which means a better prediction performance.

Figure 9 offers a comparative evaluation of all the examined IMs. The novel IM₉ is the closest to the origin of the coordinates, i.e., the HR of IM₉ is the smallest (

Table 2. Analysis parameters of different IMs.

IM	WHS	RS	RO	WHS/RS	WHS/RO	HR	WSAS
IM1	4.00	8.00	5.50	0.50	0.73	0.88	0.80
IM2	9.50	9.50	9.50	1.00	1.00	1.41	1.90
IM3	1.00	1.40	10.00	0.71	0.10	0.72	0.20
IM4	1.09	1.40	70.52	0.78	0.02	0.78	0.20
IM5	10.43	10.46	65.42	1.00	0.16	1.01	2.00
IM6	0.62	1.01	2.68	0.61	0.23	0.65	0.12
IM7	1.12	1.49	3.34	0.75	0.34	0.83	0.20
IM8	0.98	1.14	8.08	0.86	0.12	0.87	0.19
IM9	8.77	17.07	543.92	0.51	0.02	0.51	1.70

). Although IM₁ has the smallest W_HS/R_S (0.50), IM₉ is close to it (0.51). These results indicate that IM₉ can best identify the state transition from safe rocking to overturning compared with the other eight IMs.

4.4. Probability of Overturning

The standard analysis strip (SAS), whose width (W_SAS) is one-fifth of W_HS, is used to obtain the overturning probability within the specific IM value in Figure 8. The overturning probability P_ro can be estimated directly by Equation (13), which can be used in the subsequent overturning fragility analysis to obtain the fragility curves.

$$P_{\mathrm{ro}}=\frac{\mathrm{number} \mathrm{of} \mathrm{overturning} \mathrm{simulations}}{\mathrm{total} \mathrm{number} \mathrm{of} \mathrm{rocking} \mathrm{simulations}}$$

(13)

The lognormal cumulative distribution function is widely used to fit fragility functions in earthquake engineering because of its robustness in deriving fragility functions [72]:

$$P\left(\mathrm{Overturning}|IM=x\right)=\mathsf{\Phi }\left(\frac{\ln \left(x/\mu \right)}{\beta }\right)$$

(14)

where P is the probability that excitation with IM = x will cause the overturning of the rigid block, Φ is the standard normal cumulative distribution function, μ is the median of the fragility function (the IM level with a 50% probability of overturning), and β is the dispersion or logarithmic standard deviation. The least squares method is used to fit the overturning probability data obtained from Figure 8 to generate the overturning fragility curves. The error between the overturning probability data and the fitted overturning probability from the fragility function is:

$$error={\displaystyle \sum }_{i=1}^n{\left(P_i-P_{\mathrm{fit}}\left(IM=x_i\right)\right)}^2$$

(15)

where n is the number of the overturning probability data, P_i is the ith overturning probability and P_fit is the fitted overturning probability when the IM = x_i. The error can be minimized by adjusting β:

$$\beta =\arg \min \left(error\right)=\arg \min {\displaystyle \sum }_{i=1}^n{\left(P_i-\mathsf{\Phi }\left(\frac{\ln \left(x_i/\mu \right)}{\beta }\right)\right)}^2$$

(16)

Figure 10 shows the overturning fragility curves for different IMs obtained by fitting the overturning probability data. For comparing the performance of various IMs in estimating the overturning probability, the fitted analysis parameters are used to generate the coefficient of variation (CV) in this paper. CV is a standardized measure of the dispersion of a probability distribution defined as the ratio of the standard deviation β to the mean μ. Compared with the other eight IMs, IM₉ exhibits obvious superiority by having the smallest CV (Figure 11). Therefore, IM₉ is recommended as an IM for overturning fragility analysis.

4.5. Effect of Analysis Strip Width

From the process of an overturning fragility analysis, it can be found that the width of the analysis strip may affect the fragility curves. To examine this effect, we adjusted the width of the analysis strip (W_AS) to half and twice W_SAS, respectively. Figure 12 shows that, for all nine IMs, the overturning fragility curves obtained for different analysis strip widths are almost the same. The fragility curves have good robustness for different widths of the analysis strips. IM₉ maintains the smallest CV, which once again proves its superiority for overturning fragility analysis (

Table 3. Analysis parameters for different widths of the analysis strips.

IM	WAS = WSAS/2			WAS = WSAS			WAS = 2WSAS
	μ	β	CV (%)	μ	β	CV (%)	μ	β	CV (%)
IM1	4.60	0.32	6.96	4.77	0.32	6.71	4.68	0.28	5.98
IM2	7.65	1.48	19.35	7.97	1.72	21.59	8.19	2.22	27.12
IM3	1.39	0.12	8.63	1.36	0.14	10.32	1.40	0.13	9.27
IM4	1.28	0.11	8.58	1.27	0.14	11.05	1.32	0.13	9.82
IM5	8.01	0.40	5.00	7.43	0.48	6.46	7.76	0.51	6.57
IM6	1.25	0.07	5.59	1.27	0.07	5.52	1.27	0.09	7.10
IM7	1.74	0.10	5.73	1.73	0.12	6.92	1.72	0.12	6.97
IM8	1.11	0.11	9.94	1.09	0.14	12.88	1.06	0.26	24.57
IM9	12.80	0.17	1.33	13.15	0.15	1.14	13.28	0.15	1.13

).

5. Conclusions

This study provides deterministic and probabilistic views into the rocking-overturning responses of freestanding rigid blocks. The simulated results for four block models under excitation with various characteristics were conducted to generate the overturning spectrum. A comparative study of nine intensity measures, including one used for the first time in overturning fragility analysis was conducted to evaluate their capability to predict overturning. The overturning fragility curves were derived by the least-square fitting of a lognormal cumulative distribution. Some concluding remarks are drawn:

• An effective IM for overturning fragility analysis should include as many of the excitation characteristics and as much of the block geometry information as possible.
• A hybrid ratio, a parameter that can estimate the size of the hybrid IM strip within which both safe rocking and overturning may occur, is presented to quantitively compare the performance of various IMs in predicting overturning. A novel IM₉ (dimensionless floor displacement) had the best performance with the smallest HR.
• The novel IM₉ first used in an overturning fragility analysis performs best by significantly reducing the coefficient of variation compared with some well-known IMs. Thus, IM₉ is recommended as an IM for overturning fragility analysis.
• Different widths of analysis strips were used to generate the overturning fragility curves. The results show that the strip width only slightly affects the overturning fragility curves, thus revealing the good robustness of the analysis process.