An Intensity Measure for the Rocking Fragility Analysis of Rigid Blocks Subjected to Floor Motions

Abstract

A novel intensity measure (IM), dimensionless floor displacement, is presented for evaluating the seismic fragility of freestanding rigid blocks subjected to one-sine acceleration pulses in this paper. The rocking responses of rigid blocks are simulated using an equivalent single-degree-of-freedom (SDOF) model with a bespoke discrete damper to account for energy dissipation. The performance of various IMs is compared using simulation results for four different block models under different excitation conditions. In comparison to some well-known IMs, the proposed IM, determined by excitation magnitude and frequency as well as block geometry parameters, displays a considerably stronger correlation with the peak rotation of the rocking block. The comparative results show that effective IMs should consider not only the excitation characteristics but also the block geometric parameters. Finally, the fragility curve generated by the proposed IM performs best by significantly reducing the dispersion.

1. Introduction

After an earthquake, it may be prohibitively costly to restore the functionality of a building with severe content damage [1], thus calling for the evaluation of building content damage [2]. An inhibition of evaluating the seismic damage of building contents is that predicting the response of such objects is extremely difficult. Unanchored contents in buildings, typically considered as freestanding rigid blocks, may undergo complex motions during earthquake events, including sliding, twisting, rocking, impacting neighboring walls or other objects, and even overturning due to floor movement. Within these complex dominating modes of motion, rocking and sliding are deemed to be of the most importance. Since Shenton [3] presented criteria for the initiation of the different response modes of an unanchored rigid block, the sliding response, among one of the dominating modes, has been addressed separately at great length [4,5,6,7,8,9,10,11]. This paper focuses on the freestanding rigid blocks dominated only by rocking motion, which features a partial uplift from its base and a change in rotation center, and overturning may occur when the rocking rotation is large enough.

As a very early study in this area, Housner [12] proposed a seminal framework to evaluate the seismic response of a solitary rigid block placed on a rigid base. Following this pioneering work, investigators over the world have endeavored to evaluate the rocking response of freestanding blocks, including experimental [13,14,15,16,17,18,19,20,21,22,23,24] and numerical approaches [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39]. Early researchers have observed that a rocking motion has extremely complex dynamic characteristics and is barely “nonrepeatable” [40,41,42,43]. Even small changes in size, slenderness, or the details of ground motion might result in significant distinction in the rocking response. The reason for this phenomenon can be mainly attributed to the negative stiffness of rocking oscillators [44] and the complex variability in energy dissipation and transfer [45]. In this context, Yim [41] observed that systematic trends emerge when the rocking response is studied from a probabilistic point of view, with the excitation modeled as a random process [43]. Following the same idea, many investigators assessed the seismic performance of rigid blocks (e.g., rocking columns [46], bridges [47], and hospital contents [48]) via fragility analysis, a widely used method in earthquake engineering [49]. The rocking fragility is represented by a conditional probability that the damage measure (DM) will exceed a certain capacity limit state, given an IM value.

As the nonlinearity of the rocking oscillator is complex, widely used intensity measures proposed for elastic or elastoplastic oscillators are impracticable. Accordingly, many researchers have proposed various IMs for the fragility analysis of rocking blocks. Dimitrakopoulos and Paraskeva [50] explored different intensity measures in rocking and overturning fragility analysis. The results indicated that bivariate IMs provide superior estimations of the fragility to those adopting univariate ones, and dimensionless IMs are recommended for providing an approximate universal description of the rocking behavior. Petrone et al. [51] evaluated the efficiency of different IMs in predicting the probability that rigid blocks reach a specific damage state. In their research, dimensionless peak ground acceleration is demonstrated to be the most effective IM for small rigid blocks, whereas dimensionless peak ground velocity is the most effective one for large rigid blocks. Based on the dimensionless peak velocity, Sieber et al. [52] proposed a new IM considering the coefficient of restitution [12] of rigid blocks and obtained more universal results. However, the existing IMs present insufficiently strong correlations to the rocking response, thus causing large dispersion of the fragility curves.

To extend these studies, this paper presents a rigorous probabilistic investigation of the rocking response and assesses various IMs on their capability of describing the rocking response. The rocking responses of rigid blocks have been simulated with a reliable numerical model [53]. The motivation of this work is to propose a novel IM that considers not only the excitation characteristics but also the block geometric parameters, to gain a stronger correlation with the rocking response and less dispersion of the fragility curve. Consequently, the new IM, based on dimensionless floor displacement, is then proposed in this paper and evaluated through statistical analysis. The proposed IM exhibits a much stronger correlation with the dimensionless peak rotation of rocking blocks than the existing ones, thus greatly reducing the dispersion of the rocking fragility functions.

The rest of this paper is organized as follows. In Section 2, the rocking seismic response is simulated by adopting a discretely damped SDOF model, and the numerical model is validated using the experimental results. In Section 3, a three-dimensional rocking spectrum is derived as an extension of Zhang and Markris’s overturning acceleration spectrum [54]. In Section 4, we propose a new intensity measure and demonstrate its superiority by comparing its performance in rocking fragility analysis with some widely used intensity measures. Finally, in Section 5, some concluding remarks are drawn. The definitions of symbols used in this paper are listed in Appendix A.

2. Rocking Seismic Response Analysis

2.1. Numerical Model of the Rocking Block

A homogenous freestanding rigid block with a width of 2b and a height of 2h is illustrated in Figure 1. The base surface is assumed to be horizontal, rigid, and rough enough so that rocking is the only dominating mode of motion. The total mass of the block is m, and the center of mass (CM) is also its center of geometric. Equivalently, the geometry of the block can be represented by a size parameter, $R=\sqrt{b^2+h^2}$, which is the distance from CM to the pivot point, and a slenderness parameter, $\alpha =\mathrm{atan}\left(b/h\right)$. Its moment of inertia about the pivot point O or O’ is $I_{\mathrm{O}}=\frac{4}{3}m{R}^2$.

The rocking motion of the block can be fully described by the rotation θ around the pivot point. The moment equilibrium about the pivot point gives the equation of motion of an undamped rocking block under a horizontal excitation ${\ddot{u}}_0$:

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

(1)

where g is the gravity acceleration, H(θ) and B(θ) are the vertical and horizontal transient distances between CM and the current pivot point, 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.

The block will uplift and commence rocking when the horizontal excitation acceleration ${\ddot{u}}_0$ exceeds a minimum magnitude gtanα (Equation (4)). Once the rigid block starts to rock from the initial position, the restoring moment M decreases monotonically with the increase in the rotation θ, and reaches zero when θ = α, as shown by the light solid line in Figure 2. The M-θ relationship can be expressed in Equation (5). The maximum restoring moment at the initial position ($\theta =0$) is denoted as M₀ (Equation (6)).

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

(4)

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

(5)

$$M_0=mgh·\tan \alpha =mgR·\sin \alpha$$

(6)

When the rocking block switches pivot corners and impacts the floor, there is some kinetic energy lost. The energy dissipation due to impact is commonly modeled by the restitution coefficient e, which is the ratio of the relative velocity of the rocking objects after and before the collision [55]. In this case, $e={\dot{\theta }}_2/{\dot{\theta }}_1$, where ${\dot{\theta }}_1$ and ${\dot{\theta }}_2$ are the angular velocities before and after the block impacts with the floor. Housner [12] derived a rigid-body restitution coefficient e_R, represented by the slenderness parameter α of the block (Equation (7)), by conserving angular momentum before and immediately after the impact when the pivot point shifts from O to O’. The restitution coefficient e of a real-world collision also depends on the localized nonlinearity of the colliding materials and, therefore, is usually smaller than e_R [56].

$$e_R=\frac{I_O-2m{R}^2{\sin }^2\alpha }{I_O}=1-\frac{3}{2}{\sin }^2\alpha$$

(7)

In this paper, an equivalent lumped mass single-degree-of-freedom (SDOF) model is adopted to simulate the rocking response of freestanding blocks [57]. The lumped mass is modeled to be supported by a rigid link above the floor with a nonlinear elastic rotational spring at the base (Figure 3).

Considering a typical rocking block with geometry as illustrated in Figure 3a, the equation of motion of its equivalent SDOF oscillator (Figure 3b) subjected to a floor motion ${\ddot{u}}_0$ can be written as follows:

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

(8)

where $I_{\mathrm{O}}=\frac{4}{3}m{R}^2$ is the moment of inertia about the pivot point. $f_d$ is the damping force to dissipate energy during rocking; k is the tangent stiffness to model the M-θ relationship of the rocking block. Since the moment of inertia of the lumped mass m is only $m{R}^2$, an additional moment of inertia $I_{\mathrm{CM}}=\frac{m{R}^2}{3}$ is added in the model so that the total moment of inertia of the equivalent SDOF oscillator about the pivot point equals that of the original rigid block. The nonlinear M-θ relationship of the rocking block is embedded in the zero-length rotational spring at the bottom of the rigid link (Figure 3b). The nonlinear descending branch of the M-θ relationship is simplified to a linear relationship in Equation (9) (dashed line in Figure 2). Thus, the M-θ relationship embedded in the model is simplified as a bilinear elastic relationship with an initial stiffness $k_0$ within a small range, and a negative stiffness k_r beyond this range. $k_0=n\left|k_{\mathrm{r}}\right|$ is used to approximate the infinite stiffness before the rigid block starts to rock, where n is a large number. The system is assumed to oscillate linearly within the small range of ±δα on both sides of the position θ = 0, where δ = 1/(n+1).

$$M=mgR\sin \alpha ·\left(1-\frac{\theta }{\alpha }\right)=M_0\left(1-\frac{\theta }{\alpha }\right)$$

(9)

$$k\left(\theta \right)=\{\begin{array}{c}\hfill k_0=n\left|k_r\right|, \left|\theta \right|/\alpha \le \delta \\ \hfill k_r=-M_0/\alpha , \left|\theta \right|/\alpha >\delta \end{array}$$

(10)

A damping force $f_d\left(\dot{\theta }\right)$ is introduced to the simplified SDOF model to approximately account for the energy dissipation during rocking. Unlike the continuous damping model commonly used [44,58,59], a discrete damping model proposed by Liu et al. [53] is adopted to correctly simulate the energy dissipation. In this model, a viscous damping force $f_d=c_D\dot{\theta }$ applies only within the small range of ±δα on both sides of the original position θ = 0; when $\theta$ goes out of this range, the damping force equals 0; thus, the energy is dissipated if and only if the system passes its original position during rocking (Equation (11))

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

(11)

where $c_D$ is the discrete viscous damping coefficient. The coefficient $c_D$ is not constant, but proportional to the angular velocity before each impact (Figure 4), which is physically associated with the restitution coefficient during impact by the conservation of angular momentum ${\dot{\theta }}_1$ (Equation (12)). The restitution coefficient e = 0.95e_R is adopted to consider the energy dissipation of a real-world collision. The numerical simulation is performed in OpenSees [60], and the technical details can be found in our former paper [53].

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

(12)

2.2. Experimental Verifications

To demonstrate the superior performance of the discrete damping model, the simulated results are compared with the experimental results from Nasi [61,62,63] (Figure 5). Six selected runs of simulated response histories are compared with experimental results in Figure 6. The accuracy and the applicability of this model have been demonstrated by correctly approximating the maximum rotation angle and successfully estimating the occurrence of overturning [53].

3. Rocking Spectra

Zhang and Makris [54] proposed an overturning acceleration spectrum for freestanding rigid blocks subjected to one-sine acceleration pulses. The two axes of the spectrum are dimensionless pulse frequency (ω_P/P) and dimensionless peak pulse acceleration (PFA/gtanα), where T_p and ω_P = 2π/T_P are the period and circular frequency of the pulse excitation, respectively. PFA is peak floor acceleration, and $P=\sqrt{3g/4R}$ is the frequency parameter of the rigid block proposed by Housner [12]. Although the coordinate plane in the literature [54] is divided into three zones of overturning, without impact, overturning with impact, and no overturning, to focus on overturning probability, the overturning acceleration spectrum in this study is divided simply into the overturning zone and safe zone.

The overturning acceleration spectrum is evaluated by simulating the overturning responses of the four rigid block models (Figure 7) by the simplified SDOF model with discrete damping. The geometry parameters of the four models are listed in

Table 1. Geometry parameters of rigid block models.

	2b (m)	2h (m)	R (m)	α	P
Model 1	0.3785	0.9462	0.5095	0.3805	3.7981
Model 2	0.1999	0.9993	0.5095	0.1974	3.7981
Model 3	0.6971	1.7427	0.9385	0.3805	2.7986
Model 4	0.3681	1.8405	0.9385	0.1974	2.7986

, which are common sizes of objects around us. With each model excited by 100 pulses, 400 uniformly distributed cases are obtained in an overturning acceleration spectrum by adjusting the PFA and ω_P of one-sine pulse motions (Figure 8a). There is a clear boundary between the overturning zone and the safe zone in the overturning acceleration spectrum obtained from one-sine pulse motions (Figure 8b), which is also in line with the results of Zhang and Makris [54].

Three-dimensional rocking rotation spectrum can be obtained by extending the overturning acceleration spectrum with the third axe being the normalized peak rocking rotation |θ_max|/α, as shown in Figure 9. It can be observed that the peak rocking rotation is not only related to ω_P/P but also PFA/gtanα, which are usually used as intensity measures. However, it is obvious that either one is one-sided for rocking fragility analysis, which leads to the superiority of bivariate IMs [50]. An IM, for rocking fragility analysis, should be defined not only by the excitation characteristics (magnitude PFA, frequency ω_P) but also by the geometric parameters of the rigid block (size parameter R, slenderness parameter α).

4. Rocking Fragility Analysis

The rocking fragility of the freestanding rigid blocks can be expressed as the conditional probability P_f that a damage measure (DM) will exceed a certain capacity limit state (LS), given an IM value:

$$P_f=P\left(DM>LS|IM\right)$$

(13)

The probability tree diagram that incorporates the peculiarities of the rocking response and facilitates the calculation of conditional probability P_f is depicted in Figure 10. P_nr denotes the probability that the rigid block will remain resting on the ground (non-rocking response) throughout the excitation. This case corresponds to the fact that the block does not rock unless the acceleration ${\ddot{u}}_0$ exceeds the minimum threshold in Equation (5). P_ro denotes the rocking–overturning probability. The probability P_f that the DM will exceed a certain capacity limit LS given an IM value is derived by the union of two likelihoods (Figure 10), namely, the probability P_ro of overturning caused by rocking and the probability P_ex that the DM will exceed the threshold LS during rocking response without the occurrence of overturning. This paper focuses on the calculation and analysis of the latter, i.e., the probability P_ex that the DM will exceed the threshold LS during rocking response without overturning (safe rocking), and the performance of different IMs have been compared in this analysis process.

$$P_f=P_{ro}+\left(1-P_{ro}\right)P_{ex}\left(DM>LS|IM\right)$$

(14)

4.1. Damage Measure and Limit States

Dimensionless DM has been widely used because it is straightforward to evaluate the degree of rocking response [50,51,52]. For the purposes of the subsequent fragility analysis, the absolute peak rocking rotation |θ_max| normalized by the slenderness angle α is used as the DM in this paper (Equation (16)). This dimensionless DM highlights its clear physical meaning: a larger-than-0 value corresponds to the rigid block commencing rocking, whereas higher values indicate that the block experiences more severe rocking. Three apposite performance levels are proposed to assess the vulnerability of a rocking block: LS₁ = 0.1 marks observable rocking during seismic excitation, LS₂ = 0.5 indicates moderate rocking response, and LS₃ = 1.0 corresponds to extremely severe rocking. The dimensionless absolute peak rocking rotation |θ_max|/α is regularly used to judge whether the blocks are overturned or not, with greater-than-1.0 values denoting overturning [52,64]. However, this viewpoint is deemed to be controversial because a few studies have also pointed out that it is still possible for |θ_max| to exceed α without overturning [54]. Moreover, the fragility analysis results obtained by the data of safe rocking are based on the premise that no overturning occurs. That is to say, a high value of the DM (even DM >1.0) merely means that the rigid block may rock violently, and it is very likely to return to its original configuration eventually.

$$DM=\frac{\left|{\theta }_{\max }\right|}{\alpha }$$

(15)

4.2. Intensity Measures

As stated above, discovering appropriate IMs for rocking fragility analysis has been a pending challenge for a long time. A summary and examination of eight commonly used dimensionless IMs are presented in this paper, along with a comparison with the proposed IM. IM₁, IM₂, and IM₃ are dimensionless floor motion frequency, dimensionless peak floor acceleration, and dimensionless peak floor velocity, respectively:

$$I{M}_1=\frac{{\omega }_P}{P}, I{M}_2=\frac{PFA}{g\tan \alpha }, I{M}_3=\frac{P·PFV}{g\tan \alpha }$$

(16)

where PFV is the peak velocity of the pulse and $P=\sqrt{3g/4R}$ is the frequency parameter [12]. Then, IM₄, IM₅, IM₆, and IM₇ are the four bivariate IMs proposed by Dimitrakopoulos and Paraskeva [50], of which the first two are often used for rocking fragility analysis and the last two for overturning fragility analysis:

$$I{M}_4=1.484{\left(\frac{PFA}{g\tan \alpha }\right)}^{1.644}{\left(\frac{{\omega }_P}{P}\right)}^{-2.013}$$

(17)

$$I{M}_5=0.063{\left(\frac{PFA}{g\tan \alpha }\right)}^{2.954}{\left(\frac{{\omega }_P}{P}\right)}^{-0.942}$$

(18)

$$I{M}_6={\left(\frac{PFA}{g\tan \alpha }\right)}^{0.52}{\left(\frac{{\omega }_P}{P}\right)}^{-0.48}$$

(19)

$$I{M}_7={\left(\frac{PFA}{g\tan \alpha }\right)}^{0.6}{\left(\frac{{\omega }_P}{P}\right)}^{-0.4}$$

(20)

IM₈ is a newly proposed IM based on the dimensionless peak velocity that takes into account the restitution coefficient e_R (Equation (7)). This IM has been tested extensively and has been shown to produce universal results in the literature [52].

$$I{M}_8=\frac{e_R{}^4·P·PFV}{g\tan \alpha }$$

(21)

Following the same idea of dimensionless IM, we propose a new IM (i.e., IM₉) in this study. IM₉ explicitly includes excitation characteristics (magnitude PFA and frequency ω_P) and geometric parameters of the rigid block (size parameter R and slenderness parameter α). The proposed IM₉, which can be regarded as a dimensionless displacement intensity measure, has been compared with the eight IMs mentioned above in the subsequent rocking fragility analysis.

$$I{M}_9=\frac{PFA·T_P{}^2}{R\tan \alpha }$$

(22)

4.3. Probability of Limit State Exceedance during Safe Rocking

Assuming that the DM and IM are random variables following lognormal distributions, the conditional probability P_ex that an excitation with IM = x will cause the damage exceedance of a capacity limit LS during safe rocking can be written as follows:

$$P_{ex}=P_{ex}\left(DM>LS|IM=x\right)=1-\mathsf{\Phi }\left(\frac{\ln \left(LS\right)-\mu \left(x\right)}{\beta }\right)$$

(23)

where Φ is the standard (i.e., with mean 0 and standard deviation 1) normal cumulative distribution function, μ is the median value of natural logarithm of x (lnx), and β is the dispersion, or logarithmic standard deviation.

Assume there is a linear relationship between μ and ln(IM)

$$\mu =a+b\ln \left(IM\right)$$

(24)

which is a typical trick that facilitates the estimation of parameters a and b through linear regression analysis (Figure 11). Additionally, the dispersion β can be obtained by Equation (25). According to Equation (23), given a capacity limit LS, the corresponding fragility curves of the freestanding rigid blocks during safe rocking can be obtained. A lower value of β means less dispersion of the demand and, consequently, a more efficient IM.

$$\beta =\sqrt{\frac{1}{n-1}{\displaystyle \sum }_{i=1}^n{\left(\ln D{M}_i-\mu \left(x_i\right)\right)}^2}$$

(25)

Figure 11 presents the linear regression results of the DM with respect to different IMs in logarithmic space, considering only the cases of safe rocking. The fitting parameters a, b, and dispersion β are shown in

Table 2. Linear regression analysis parameters of different IMs.

IM	a	b	β	R2
IM1	1.1730	−1.0999	0.7083	0.1846
IM2	−2.1786	0.8195	0.5597	0.4908
IM3	−0.3761	1.1643	0.2569	0.8928
IM4	−0.1313	0.7151	0.1827	0.9457
IM5	−0.8771	0.3274	0.4765	0.6310
IM6	−0.5569	2.2191	0.2829	0.8699
IM7	−1.1339	1.8261	0.3689	0.7788
IM8	−0.0879	0.7671	0.3782	0.7675
IM9	−2.5232	1.0484	0.1113	0.9799

. The coefficient of determination R², which is used to evaluate the efficiency of the regression, is also included. A closer-to-1 R² value indicated better goodness of fit. Among the commonly used univariate IMs, dimensionless peak floor velocity IM₃ performs the best, with a smaller β and larger R². This is consistent with previous research results [22,51,64]. Compared with univariate IMs, the four bivariate IMs proposed by Dimitrakopoulos and Paraskeva [50] generally produce better results overall, with IM₄ performing particularly well. The new IM₉ proposed in this paper exhibits a much stronger correlation with the DM in logarithmic space than all the existing IMs examined in this paper, with the smallest β and the largest R². Therefore, we recommend using IM₉ as an intensity measure for rocking fragility analysis.

The rocking fragility curves obtained by various IMs according to the three performance thresholds mentioned above are shown in Figure 12. The proposed IM₉, with the smallest dispersion β, consistently shows the steepest curve. The best-performing fragility curves have been obtained with respect to the most effective IM₉, which can conveniently estimate the probability P_ex that an excitation will cause the exceedance of a performance limit during safe rocking.

5. Conclusions

This study examined the seismic behaviors of the freestanding rigid blocks subjected to one-sine acceleration pulses. We simulated four blocks, in common sizes of objects around us, under excitation with different amplitudes and different frequencies using a reliable numerical model. The seismic rocking fragility has been assessed within a probabilistic framework. Eight well-established intensity measures, along with a new intensity measure proposed in this paper, were evaluated on their capability to describe the excitation-induced peak rocking rotation. With the cases of safe rocking solely studied here, the fragility curves were derived and approximated by fitting lognormal cumulative distributions. The following conclusions can be drawn from the results:

• An effective IM should take into account not only the excitation characteristics (magnitude PFA, frequency ω_P) but also the geometric parameters of the rigid blocks (size parameter R, slenderness parameter α);
• The dimensionless peak floor velocity performs better among the univariate IMs commonly used in rocking fragility analysis. Bivariate IMs perform better overall, but require more computation;
• A novel IM explicitly including excitation characteristics and geometric parameters of the rigid blocks is proposed in this paper. The proposed IM exhibits a much stronger correlation with the DM in logarithmic space; consequently, the proposed IM yields the smallest β in linear regression analysis, which results in the best-performing fragility curves;
• Future studies should aim at evaluating the overturning fragility, as well as the rocking behavior subject to excitations in the real world.