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

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Rocking Seismic Response Analysis

#### 2.1. Numerical Model of the Rocking Block

_{0}(Equation (6)).

_{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].

_{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).

_{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].

#### 2.2. Experimental Verifications

## 3. Rocking Spectra

_{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.

_{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].

_{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

_{f}that a damage measure (DM) will exceed a certain capacity limit state (LS), given an IM value:

_{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.

#### 4.1. Damage Measure and Limit States

_{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: LS

_{1}= 0.1 marks observable rocking during seismic excitation, LS

_{2}= 0.5 indicates moderate rocking response, and LS

_{3}= 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.

#### 4.2. Intensity Measures

_{1}, IM

_{2}, and IM

_{3}are dimensionless floor motion frequency, dimensionless peak floor acceleration, and dimensionless peak floor velocity, respectively:

_{4}, IM

_{5}, IM

_{6}, and IM

_{7}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:

_{8}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].

_{9}) in this study. IM

_{9}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 IM

_{9}, 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.

#### 4.3. Probability of Limit State Exceedance during Safe Rocking

_{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:

^{2}, which is used to evaluate the efficiency of the regression, is also included. A closer-to-1 R

^{2}value indicated better goodness of fit. Among the commonly used univariate IMs, dimensionless peak floor velocity IM

_{3}performs the best, with a smaller β and larger R

^{2}. 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 IM

_{4}performing particularly well. The new IM

_{9}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 R

^{2}. Therefore, we recommend using IM

_{9}as an intensity measure for rocking fragility analysis.

_{9}, with the smallest dispersion β, consistently shows the steepest curve. The best-performing fragility curves have been obtained with respect to the most effective IM

_{9}, which can conveniently estimate the probability P

_{ex}that an excitation will cause the exceedance of a performance limit during safe rocking.

## 5. Conclusions

- 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.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Symbol | Definition |
---|---|

2b | Width |

2h | Height |

$R$ | Size parameter |

$\alpha $ | Slenderness parameter |

${I}_{\mathrm{O}}$ | Moment of inertia |

$\theta $ and $\ddot{\theta}$ | Rotation angle and rotational angular acceleration |

${\ddot{u}}_{0}$ | Horizontal excitation |

$H$ and $B$ | Vertical and horizontal transient distances |

g | Gravity acceleration |

M | Restoring moment |

M_{0} | Maximum restoring moment |

${\dot{\theta}}_{1}$ and ${\dot{\theta}}_{2}$ | Angular velocities before and after impacts |

$e$ | Restitution coefficient |

${e}_{R}$ | Rigid-body restitution coefficient |

${f}_{d}$ | Damping force |

$k$ | Tangent stiffness |

${I}_{CM}$ | Additional moment of inertia |

${k}_{0}$ | Initial stiffness |

${k}_{r}$ | Negative stiffness |

$n$ | A large number |

δα | Small range around initial position |

${c}_{D}$ | Discrete viscous damping coefficient |

θ_{max} | Peak rocking rotation |

T
_{p} | Period of pulse excitation |

ω_{P} | Circular frequency of pulse excitation |

P | Block frequency parameter |

PFA | Peak floor acceleration |

PFV | Peak floor velocity |

IM | Intensity measure |

DM | Damage measure |

LS | Limit state |

${P}_{f}$ | Conditional probability |

${P}_{ro}$ | Overturning probability |

${P}_{ex}$ | Probability for DM exceeding LS within safe rocking |

x | IM value |

μ | Median value of $\mathrm{ln}x$ |

a and b | Linear regression parameters |

β | Dispersion |

R^{2} | Coefficient of determination |

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**Figure 3.**(

**a**) Lumped mass representation of rigid block and (

**b**) equivalent SDOF model of a rigid rocking block.

**Figure 4.**Hysteretic curves of a free-rocking SDOF system of α = 0.2 and R = 0.38 m: (

**a**) total resisting moment with discrete damping, (

**b**) discrete damping force.

**Figure 5.**Experimental set up (source: PURR—Stability_of_Rocking_Structures_20170825 (purdue.edu), accessed on 24 December 2022).

**Figure 6.**Comparison of rotation histories of non-overturning (

**a**–

**c**) and overturned (

**d**–

**f**) runs by SDOF models.

**Figure 11.**Linear regression analysis of the maximum normalized response with respect to different IMs: (

**a**–

**h**) IM

_{1}–IM

_{8}; (

**i**) the proposed IM

_{9}.

**Figure 12.**Rocking fragility curves for different IMs: (

**a**–

**h**) IM

_{1}–IM

_{8}; (

**i**) the proposed IM

_{9}.

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 |

IM | a | b | β | R^{2} |
---|---|---|---|---|

IM_{1} | 1.1730 | −1.0999 | 0.7083 | 0.1846 |

IM_{2} | −2.1786 | 0.8195 | 0.5597 | 0.4908 |

IM_{3} | −0.3761 | 1.1643 | 0.2569 | 0.8928 |

IM_{4} | −0.1313 | 0.7151 | 0.1827 | 0.9457 |

IM_{5} | −0.8771 | 0.3274 | 0.4765 | 0.6310 |

IM_{6} | −0.5569 | 2.2191 | 0.2829 | 0.8699 |

IM_{7} | −1.1339 | 1.8261 | 0.3689 | 0.7788 |

IM_{8} | −0.0879 | 0.7671 | 0.3782 | 0.7675 |

IM_{9} | −2.5232 | 1.0484 | 0.1113 | 0.9799 |

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**MDPI and ACS Style**

Liu, H.; Huang, Y.; Liu, X.
An Intensity Measure for the Rocking Fragility Analysis of Rigid Blocks Subjected to Floor Motions. *Sustainability* **2023**, *15*, 2418.
https://doi.org/10.3390/su15032418

**AMA Style**

Liu H, Huang Y, Liu X.
An Intensity Measure for the Rocking Fragility Analysis of Rigid Blocks Subjected to Floor Motions. *Sustainability*. 2023; 15(3):2418.
https://doi.org/10.3390/su15032418

**Chicago/Turabian Style**

Liu, Hanquan, Yong Huang, and Xiaohui Liu.
2023. "An Intensity Measure for the Rocking Fragility Analysis of Rigid Blocks Subjected to Floor Motions" *Sustainability* 15, no. 3: 2418.
https://doi.org/10.3390/su15032418