# Rocking Blocks Stability under Critical Pulses from Near-Fault Earthquakes Using a Novel Energy Based Approach

^{*}

## Abstract

**:**

## Featured Application

**A novel energy based approach is derived from basic physical principles to assess the stability of rocking blocks and inverted pendulum structures subjected to single pulses of various shapes. The Discrete Element Method is used to numerically verify the theoretical results. The normalized stability chart can be used to identify the critical pulses from near-fault earthquakes and study the stability of various objects prone to overturning once the corresponding intensity measures are derived. The proposed approach offers a robust lower bound for the body of work published in the acceleration–frequency spectrum. The results can also be used to characterize the critical acceleration–period content of historical earthquakes based on archaeoseismological evidence.**

## Abstract

## 1. Introduction

#### 1.1. Theoretical and Numerical Investigations of Rocking Blocks Stability

#### 1.2. Experimental and Empirical Investigations of Rocking Blocks Stability

#### 1.3. Representation of the Dominant Pulse of Near-Fault Earthquakes

#### 1.4. Archaeoseimological Relevance

#### 1.5. The Need for Efficient Scalar Intensity Measures

## 2. Theoretical Model

_{p}, and (v) the shape of the pulse. With reference to Figure 1 adopted from Housner [1]: the block height is 2 h, its width is 2 b, its mass is m, its weight is W = mg with g being the acceleration of gravity and its rotational moment of inertia is I

_{o}= (4/3) mR

^{2}; with R = √(h

^{2}+ b

^{2}). As the block rotates around its corner O, the block rotation, θ, increases from zero to a limit value at which the block overturns. The limit value of θ is the angle α = tan

^{−1}(b/h). Values of (b/h) are relatively small for slender blocks and α ≈ b/h.

_{O}= (ma)h, must exceed the restoring moment of the block, M

_{R}= (mg)b. Hence for inception of rocking it is necessary that M

_{O}≥ M

_{R}or the horizontal acceleration caused by the applied pulse, a, exceeds at some point the minimum value of αg:

^{2}/2 and cos

^{2}α = 1, hence:

^{2}= WR/I

_{o}, or p = $\sqrt{\frac{3g}{4R}}$

_{p}= duration of pulse, a

_{p}= peak acceleration of the pulse, and β = fullness factor according to the above equation [19,22,68]. Therefore, the critical impulse curve or relationship is given by:

_{p}= αg, is at ${\overline{t}}_{p}$:

_{p}and t

_{p}is then given by: ${\overline{a}}_{p}=\frac{{a}_{p}}{\alpha g}$, where:

_{p}, we have:

^{0.5}smaller.

## 3. Numerical Verification

_{n}; a spring in the shear direction with stiffness = K

_{s}; and a Coulomb slip model.

- (a)
- For periods greater roughly than 0.4 s for the 0.5 m block and 0.6 s for the 1 m block, the base and the top of the block move together. Inception of rocking is indicated by a minor lag between the two displacements followed by rapid increase over one or two impacts leading to overturning.
- (b)
- For periods smaller than 0.4 s for the 0.5 m block and 0.6 s for the 1 m block, multiple rocking takes place followed by overturning. The rocking response exhibits an apparent period that is different from the period of the applied excitation with an increase in the magnitude of rocking, akin to resonance, which leads to overturning after a few cycles.

^{0.5}. N = 1 for the 20 m block and N = 20 and 40 for the 1 m and 0.5 m blocks, respectively.

## 4. Discussion

^{−1}(b/h) is the critical stability angle and αg is the minimum theoretically required acceleration for overturning.

_{p}/p in the interval of (0; 2] and the domain of the small periods (2.32 s; 0.68 s) corresponds to ω

_{p}/p in the interval of (2; 6.8). Similarly to Figure 8, Figure 9 compares the original numerical data of Spanos and Koh [8] for a block with tan α = 0.25 under a harmonic acceleration forcing function to the predicted safe–unsafe boundary for half sine impulse under the same maximum friction factor limitation.

_{p}= a

_{p}T

_{p}

^{2}, of the ground excitation as an intensity measure for the near-fault single pulse noting that they still needed one more size related variable to properly characterize the rocking instability. Kavvadias et al. [91] investigated the effect of 23 ground motion intensity measures (IMs) on the stability of ancient rocking columns trying to identify the most relevant ones. They found that acceleration based parameters could predict rocking response only if combined with a frequency content parameter and that free-standing columns exhibited increased stability due to increasing size and rocking behavior. They only collapsed under long period extreme seismic records. Pappas et al. [92] investigated the efficiency of using Peak Ground Acceleration (PGA) and Peak Ground Velocity (PGV) as earthquake IM for the seismic vulnerability assessment of monolithic rocking bodies. To do so, a three-dimensional model of an existing free-standing ancient column of the Roman Agora of Thessaloniki was analyzed numerically with a three-dimensional DEM. A strong influence of the mean frequency of the ground motions, fm, on the structural response was observed and the authors concluded that it is deemed more efficient to consider dual Intensity Measures such as, (PGA, fm) and (PGV, fm) pairs, instead of single PGA and PGV amplitudes. Some of their numerical results can be interpreted and proved using the simple energy based critical pulse approach presented above. Giouvanidis and Dimitrakopoulos [23] showed that duration-based IMs outperformed other examined scalar IMs such as intensity, frequency, and/or energy but would need to be used concurrently with other IMs in order to fully predict the rocking block response. The works of Kavvadias et al. [90], Pappas et al. [91], and Giouvanidis and Dimitrakopoulos [23] provide solid support for the use of to the proposed simple energy based critical pulse approach as a ground motion intensity measure. The intensity measure would be constituted of two measures one shape related and the other energy related, noting that the three independent variables used in the theoretical derivation are block slenderness, α, block size, R, and pulse shape, β. Purvance et al. [31] identified the PGV/PGA as a reliable IM to predict the PGA at first overturning as it is directly correlated with the predominant acceleration pulse of various waveforms including five near-fault earthquake records along with synthetic waveforms. In fact the ratio PGV/PGA returns a time value that is akin to the duration, t

_{p}, of a step pulse with β = 1. Plots a, c, and d of Figure 7 in [31] present PGA versus PGV/PGA shaking table results on symmetrical 1.2 m high columns with α = 0.29, 0.21, and 0.27 respectively. The PGA data was normalized by αg and the PGV/PGA data was normalized by 1/p for all three columns and plotted in Figure 10 against the safe–unsafe boundary for half step impulse. The normalized experimental data points overlap as predicted by the theory and while they follow the same trend as the theoretical model they fall below its prediction for the same reasons presented above in the discussion of Figure 7 presenting the experimental results of the authors [79].

## 5. Conclusions

^{0.5}.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Rocking Block after Housner [1].

**Figure 2.**(

**a**) Safe–unsafe boundary for half sine impulse signal and (

**b**) shape of critical half-sine impulses.

**Figure 3.**(

**a**) Schematic of the numerical model. (

**b**) Calculation cycle steps for the distinct element method.

**Figure 7.**Experimental results for 0.5 m solid cylinder compared to theoretical and UDEC model for 0.5 m block after [79].

**Figure 8.**Safe–unsafe boundary for half sine impulse drawn on top of the overturning acceleration spectrum of the freestanding rocking column with tan α = 0.25 when it is subjected to a one-sine acceleration pulse after Makris and Kampas [82].

T (s) | Velocity Amplitude (m/s) | Acceleration Amplitude (g) | Block, 0.5 m, 25 cycles, Displacement History (Base and Top) |
---|---|---|---|

0.2 | 0.091 | 0.2914 | |

0.3 | 0.086 | 0.1836 | |

0.4 | 0.065 | 0.1041 | |

0.6 | 0.096 | 0.1025 | |

0.9 | 0.144 | 0.1025 | |

1.5 | 0.243 | 0.1038 |

T (s) | Velocity Amplitude (m/s) | Acceleration Amplitude (g) | Block, 1 m, 25 cycles, Displacement History (Base and Top) |
---|---|---|---|

0.3 | 0.127 | 0.2711 | |

0.4 | 0.116 | 0.1857 | |

0.5 | 0.092 | 0.1178 | |

0.6 | 0.097 | 0.1035 | |

1.0 | 0.164 | 0.1050 | |

1.5 | 0.248 | 0.1059 |

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**MDPI and ACS Style**

Karam, G.; Tabbara, M.
Rocking Blocks Stability under Critical Pulses from Near-Fault Earthquakes Using a Novel Energy Based Approach. *Appl. Sci.* **2020**, *10*, 5924.
https://doi.org/10.3390/app10175924

**AMA Style**

Karam G, Tabbara M.
Rocking Blocks Stability under Critical Pulses from Near-Fault Earthquakes Using a Novel Energy Based Approach. *Applied Sciences*. 2020; 10(17):5924.
https://doi.org/10.3390/app10175924

**Chicago/Turabian Style**

Karam, Gebran, and Mazen Tabbara.
2020. "Rocking Blocks Stability under Critical Pulses from Near-Fault Earthquakes Using a Novel Energy Based Approach" *Applied Sciences* 10, no. 17: 5924.
https://doi.org/10.3390/app10175924