# Stochastic Dynamic Analysis of Cultural Heritage Towers up to Collapse

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Properties of the Case-Study Towers

#### 2.1. Selection of Case-Study Towers

^{2}) amongst the case-study towers and was first erected in 1534 A.D. A characteristic of this tower is the presence of a bi-part foundation structure forming an arch and serving as an entrance gate.

^{2}and very few openings.

^{2}and a height of 19.50 m. The walls are made of stone masonry. Internal levels are made of timber beams and planks forming five storeys (Figure 1g). Tie rods exist at the upper two levels which host the bells. The hip roof is also wooden.

#### 2.2. Material Properties of the Towers

^{2}being realistic assumptions [23,32,33,34].

## 3. Failure Mechanisms

_{s}and the main system T

_{1}to estimate the filtering effect of the underlying structure is given by the next expression [43]:

_{s}and ξ

_{1}are the modal dampings of the rocking and the main system, respectively. For purely rocking systems, the damping was estimated to equal 3% [38]. As here friction is present, the damping is assumed to be 5%, equal to that of typical masonry buildings [44] and, therefore, their ratio ξ

_{s}/ ξ

_{1}is equal to 1. A reasonable approximation of the fundamental mode of masonry in regular buildings is given by the ratio $\left(Z/h\right)$, in which $h$ stands for the height of the structure measured and $Z$ stands for the elevation of the centre of gravity of the mechanism. In Equation (5), ${\gamma}_{1}$ is the corresponding modal participation factor which depends on the number of internal floors $n$ [45] as ${\gamma}_{1}=3n/\left(2n+1\right)$. The results of the analysis are presented in Section 6 with those from FE simulations.

## 4. Rocking Response

_{0}the polar moment of inertia, and the tangent of α is the ratio b/h. The weight from the upper structure is P while the horizontal forces are assumed inertial forces and thus, connected to W and P by the coefficient λ. The response can be given in terms of displacements δx and δy, or in terms of rotation θ. Angle β and radius R are defined from the points of rotation and the centre of mass as shown in Figure 3.

_{0})

^{1/2}. The solution of this 2nd order non-homogeneous differential equation is given by:

_{0}is the initial rotation. The error introduced by approximating Equation (6) through Equation (7) and its exact solution given by Equation (8) is small (about 2%). By assuming θ = 0, i.e., passing through the equilibrium point, then the time will be t = T/4 where T is the period of the oscillation as a total period would be the time required to return to the initial inclined position. The period vs. the ratio θ/α is plotted in Figure 4 where it is seen that the rocking response varies with θ/α—showing a dependence on the geometric properties (α = b/h)—as well as with θ.

## 5. Variability Analysis

#### 5.1. Variations in Material Properties

_{k}is evaluated applying the Eurocode 6 empirical formula [44] and using the coefficients suggested by [61] for units from stone masonry:

_{m}and units f

_{k}strengths), a parametric analysis was conducted regarding a generic tower whose model is shown in Figure 7a. The tower was assumed to be founded on rocky soil and its footprint is a square 4.5 × 4.5 m

^{2}while its height is 27.5 m. The tower was finely discretised in Abaqus [47] with square plane elements 0.3 × 0.3 m

^{2}fixed at the base. The material properties of masonry are assumed to be linear; the interaction between the cracked parts at the belfry is ruled by a friction coefficient equal to 0.5. The collapse mechanism is shown in Figure 7b. The influence of the material properties is illustrated in Figure 8 in terms of maximum horizontal displacement at the control point (top) vs. the homogenised (effective) elastic modulus: for an increase 50% in the elastic modulus the maximum top displacement is only changed by approximately 25%.

#### 5.2. Variations in Seismic Excitation

## 6. Seismic Fragility of Towers

#### 6.1. FE Models

#### 6.2. Collapse Analysis

#### 6.3. Fragility Analysis

## 7. Conclusions

^{2}. The proximity of their locations and the use of the same material sources justifies the adoption of similar elastic properties for all of them, despite the differences in their construction dates. The analysis showed that small variations in the mechanical properties do not substantially affect the global response, as the rocking type of failure dominates. The basic assumption of the procedure is a two-step analysis where the collapse mechanism is first identified and then an incremental dynamic analysis (IDA) of the FE model is performed.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Case-study towers in Mount Athos: (

**a**) Caracallou, (

**b**) Koutloumousiou, (

**c**) Vatopedi, (

**d**) Philotheou, (

**e**) Protaton, (

**f**) Dionysiou (

**g**) Xenophontos, and (

**h**) Iveron.

**Figure 2.**Damage modes for towers: (

**a**) overturning, (

**b**) separation, (

**c**) diagonal failure, and (

**d**) belfry failure.

**Figure 5.**Sequence of the rocking of a block and stress field in various positions (von Mises stress field in kPa): (

**a**) inclined initial (t = 0); (

**b**) equilibrium (t = T/4); (

**c**) equilibrium-beginning of new rotation inclined opposite (t = T/4); (

**d**) opposite inclination (t = T/2); (

**e**) equilibrium (t = 3T/4); and (

**f**) initial (t = T).

**Figure 8.**Variation of the rocking max displacement at the top for varying effective material properties.

**Figure 9.**Seismic recordings in terms of spectral displacement and spectral acceleration (modal damping ζ = 5%).

**Figure 10.**FE models of the case-study towers; results for gravity loads (von Mises stress) in kPa: (

**a**) Caracallou, (

**b**) Koutloumousiou, (

**c**) Vatopedi, (

**d**) Philotheou, (

**e**) Protaton, (

**f**) Dionysiou, (

**g**) Xenophontos, and (

**h**) Iveron.

**Figure 11.**Collapse mechanism of the case-study towers (von Mises stress in kPa): (

**a**) Caracallou, (

**b**) Koutloumousiou, (

**c**) Vatopedi, (

**d**) Philotheou, (

**e**) Protaton, (

**f**) Dionysiou, (

**g**) Xenophontos, and (

**h**) Iveron.

**Figure 12.**Spectral capacity curves (in black from the IDA, in red from the limit analysis): (

**a**) Caracallou, (

**b**) Koutloumousiou, (

**c**) Vatopedi, (

**d**) Philotheou, (

**e**) Protaton, (

**f**) Dionysiou, (

**g**) Xenophontos, and (

**h**) Iveron.

**Figure 13.**The definition of the limit states on the capacity curve for Protaton bell-tower and Northridge earthquake.

**Figure 14.**Fragility curves: (

**a**) Caracallou, (

**b**) Koutloumousiou, (

**c**) Vatopedi, (

**d**) Philotheou, (

**e**) Protaton, (

**f**) Dionysiou, (

**g**) Xenophontos, and (

**h**) Iveron.

**Figure 15.**Fragility curves: (

**a**) towers, (

**b**) campaniles (small openings’ area), and (

**c**) campaniles (large openings’ area).

Tower | Total | Belfry | Wall | Area (m^{2}) | Slenderness |
---|---|---|---|---|---|

Height (m) | Thickness (m) | ||||

Caracallou | 27.75 | No | 1.52 | 81.00 | 3.08 |

Koutloumousiou | 26.90 | No | 1.15 | 28.09 | 5.08 |

Vatopaidion | 25.55 | Yes | 0.85 | 20.25 | 5.68 |

Philotheou | 24.90 | Yes | 1.05 | 30.25 | 4.53 |

Protaton | 24.00 | Yes | 0.85 | 20.25 | 5.33 |

Dionysiou | 23.16 | No | 1.25 | 49.00 | 3.31 |

Iveron | 20.52 | Yes | 1.05 | 45.00 | 3.06 |

Xenophontos | 19.50 | Yes | 1.05 | 49.00 | 2.79 |

Νο. | f_{m}(MPa) | f_{b}(MPa) | f_{k}(MPa) | E(GPa) |
---|---|---|---|---|

1 | 1 | 70 | 3.91 | 1.96 |

2 | 1 | 100 | 5.02 | 2.51 |

3 | 1 | 85 | 4.48 | 2.24 |

4 | 1 | 115 | 5.54 | 2.77 |

5 | 1 | 130 | 6.04 | 3.02 |

6 | 1.5 | 70 | 4.42 | 2.21 |

7 | 1.5 | 95 | 5.47 | 2.74 |

8 | 1.5 | 110 | 6.07 | 3.03 |

9 | 2 | 75 | 5.06 | 2.53 |

10 | 2 | 85 | 5.52 | 2.76 |

11 | 2 | 95 | 5.97 | 2.98 |

12 | 2 | 65 | 4.57 | 2.29 |

Group 1: Towers | Group 2: Campaniles (Small Openings’ Area) | Group 3: Campaniles (Large Openings’ Area) |
---|---|---|

Dionysiou | Vatopedi | Iveron |

Caracallou | Philotheou | Protaton |

Koutloumousiou | Xenophontos | - |

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**MDPI and ACS Style**

Kouris, E.-G.S.; Kouris, L.-A.S.; Konstantinidis, A.A.; Kourkoulis, S.K.; Karayannis, C.G.; Aifantis, E.C.
Stochastic Dynamic Analysis of Cultural Heritage Towers up to Collapse. *Buildings* **2021**, *11*, 296.
https://doi.org/10.3390/buildings11070296

**AMA Style**

Kouris E-GS, Kouris L-AS, Konstantinidis AA, Kourkoulis SK, Karayannis CG, Aifantis EC.
Stochastic Dynamic Analysis of Cultural Heritage Towers up to Collapse. *Buildings*. 2021; 11(7):296.
https://doi.org/10.3390/buildings11070296

**Chicago/Turabian Style**

Kouris, Emmanouil-Georgios S., Leonidas-Alexandros S. Kouris, Avraam A. Konstantinidis, Stavros K. Kourkoulis, Chris G. Karayannis, and Elias C. Aifantis.
2021. "Stochastic Dynamic Analysis of Cultural Heritage Towers up to Collapse" *Buildings* 11, no. 7: 296.
https://doi.org/10.3390/buildings11070296